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ftr V 1 UNITS AND PHYSICAL CONSTANTS. UMTS AND PHYSICAL CONSTANTS. BY J. D. EVERETT, M.A., D.C.L., F.R.S., F.R.S.E., PROFESSOR OF NATURAL PHILOSOPHY IN QUEEN'S COLLEGE, BELFAST. SECOND EDITION. MACMILLAN AND CO., AND NEW YORK. 1886. [The right of translation and reproduction is reserved. ] GLASGOW : PRINTED AT THE UNIVERSITY PRESS BY ROBERT MACLEHOSE. PREFACE TO "ILLUSTRATIONS OF THE C.G.S. SYSTEM OF UNITS," PUBLISHED BY THE PHYSICAL SOCIETY OF LONDON IN 1875. THE quantitative study of physics, and especially of the relations between different branches of physics, is every day receiving more attention. To facilitate this study, by exemplifying the use of a system of units fitted for placing such relations in the clearest light, is the main object of the present treatise. A complete account is given of the theory of units ab initio. The Centimetre-Gramme-Second (or C.G.S.) sys- tem is then explained ; and the remainder of the work is occupied with illustrations of its application to various branches of physics. As a means to this end, the most important experimental data relating to each subject are concisely presented on one uniform scale a luxury hitherto unknown to the scientific calculator. I am indebted to several friends for assistance in special departments but especially to Professor Clerk Maxwell and Professor G. C. Foster, who revised the entire manu- script of the work in its original form. Great pains have been taken to make the work correct as a book of reference. Readers who may discover any errors will greatly oblige me by pointing them out. PREFACE TO FIRST EDITION OF UNITS AND PHYSICAL CONSTANTS, 1879. THIS Book is substantially a new edition of my ' ; Illus- trations of the C.G.S. System of Units " published in 1875 by the Physical Society of London, supplemented by an extensive collection of physical data. The title has been changed with the view of rendering it more generally intelligible. Additional explanations have been given upon some points of theory, especially in connection with Stress and Strain, and with Coefficients of Diffusion. Under the former head, I have ventured to introduce the terms "resilience" and "coefficient of resilience," in order to avoid the multiplicity of meanings which have become attached to the word "elasticity." A still greater innovation has been introduced in an extended use of the symbols and processes of multiplication and division, in connection with equations which express not numerical but physical equality. The advantages of this mode of procedure are illustrated by its application to the solution of the most difficult problems on units that I have been able to collect from standard text-books (chiefly from Wormell'a ' Dynamics '). I am indebted to several friends for contributions of experimental data. viii PEEFACE TO FIRST EDITION. A Dutch translation of the first edition of this work has been made by DR. C. J. MATTHES, Secretary of the Royal Academy of Sciences of Amsterdam, and was pub- lished in that city in 1877. Though the publication is no longer officially connected with the Physical Society, the present enlarged edition is issued with the Society's full consent and approval. PREFACE TO THE PRESENT EDITION. IN collecting materials for this edition, I have gone care- fully through the Transactions and Proceedings of the Royal Society, the Royal Society of Edinburgh, and the Physical Society of London, from 1879 onwards, besides, consulting numerous papers, both English and foreign, which have been sent to me by their authors. I have also had the advantage of the co-operation of Dr. Pierre Chappuis (of the Bureau International des Poids et Mesures), who has for some years been engaged in pre- paring a German edition. Several items have been ex- tracted from the very elaborate and valuable PJiysikaliscJt- Chemische Tabellen of Landolt and Bb'rnstein (Julius Springer, Berlin, 1883). Among friends to whom I am indebted for data or useful suggestions, are Prof. Barrett, Mr. J. T. Bottomley, Prof. G. C. Foster, Prof. Lodge, Prof. Newcomb, Mr. Preece, and the Astronomer-Royal. The expository portions of the book are for the most part unchanged ; but a Supplemental Section has been added (p. 34) on physical deductions from the dimensions of units ; a simplification has been introduced in the dis- cussion of adiabatic compression (p. 125); and the account of thermoelectricity (p. 172) has been re-written and enlarged. The name "thermoelectric height" has been introduced to denote the element usually represented by the ordinates of a thermoelectric diagram. x PREFACE TO THE PRESENT EDITION. The preliminary " Tables for reducing other measures to C.G.S. measures" have been greatly extended, and in each case the reciprocal factors are given which serve for reducing from C.G.S. measures to other measures. Pro- fessor Miller's comparison of the kilogramme and pound is supplemented by three later comparisons officially made at the Bureau International. A nearly complete list of the changes and additions now introduced is appended to this Preface, as it will probably be useful to possessors of the previous edition. The adoption of the Centimetre, Gramme, and Second, as the fundamental units, by the International Congress of Electricians at Paris in 1881, led to the immediate execution of a French translation of this work, which was published at Paris by Gauthier-Yillars in 1883. The German translation was commenced about the same time, but the desire to perfect its collection of physical data has caused much delay. It will be brought out by Ambrosius Earth, the publisher of Wiedemann's Annalen. A Polish edition, by Prof. J. J. Boguski, was published at Warsaw in 1885 ; and permission has been asked and granted for the publication of an Italian edition. J. D. EVERETT. BELFAST, September, 1886. LIST OF CHANGES AND ADDITIONS. Tables for conversion to and from C.G.S., Formula for g, " Watt " defined, Physical deductions from dimensions, Specific gravity table, - Surface tension of liquids, Thickness of soap films, - Poissoii's ratio, Velocity of light, Indices of refraction of crystals, etc. , Refraction and dispersion of gases, - Rotation by quartz, Candle, carcel, etc., Specific heat, Melting, Boiling, Pressure of steam from to 150, Critical points of gases, - Conductivity (thermal) of solids, ,, ,, of liquids, Joule's equivalent, Adiabatic compression, - Expansion of mercury, Collected data for air, Density of moist air, Magnetic susceptibility, ...... Greenwich magnetic elements, - Magneto-optic rotation, - - - - Ratio of the two units of electricity, Specific inductive capacity, PAGES 1-4 26 30 - 34-37 40 - 49, 50 - 50, 51 62 76 - 80, 81 - 83-85 85 86 - 90-94 95-97 98 102 103 115, 116 117 121 125, 126 129 129 130 133 138 139 146 148-150 xii CHANGES AND ADDITIONS. PAGES Practical units, 151, 152 Resolutions of Congress and Conference, - - - 153, 154 Resistance, .... 158-164 Gauge and resistance of wires, - 165, 166 Electro-motive force of cells, - 167, 168 Thermoelectricity, - - 172-178 Electrochemical equivalents, 179-180 Heat of combination of cells, .... 181, 182 Compression of liquids, 189 Expression of decimal multiples, etc., - - - - 190 CONTENTS. PAGES Tables for Reducing to and from C.G.S. Measures, - 1-4 Chapter!. General Theory of Units, - - - 5-18 -Chapter II. Choice of Three Fundamental Units, - 19-24 .Chapter III. Mechanical Units, 25-34 Supplemental Section, on Physical Deductions from Dimensions, 34-37 Chapter IV. -Hydrostatics, - 38-51 Chapter V. Stress, Strain, and Resilience, - 52-64 -Chapter VI. Astronomy, - - - 65-69 Chapter VII. Velocity of Sound, - - 70-74 Chapter VIII. Light, - 75-86 Chapter IX. Heat, - 87-130 . Chapter X. Magnetism, - - - - 131-139 Chapter XL Electricity, - - - 140-188 Omission and Suggestion, - - - - - 189, 190 Appendix. Reports of Units Committee of British Association,- ... 191-195 Index, 196-200 UNITS AND PHYSICAL CONSTANTS. TABLES FOR REDUCING TO AND FROM C.G.S. MEASURES. The abbreviation cm. is used for centimetre or centimetres, gm. gramme or grammes, c.c. cubic centimetre(s). The numbers headed "reciprocals" are the factors for reducing from C.G.S. measures. Length. 1 inch, - 1 foot, - 1 yard,- 1 mile, 1 nautical mile, - cm. 2-5400 30-4797 91-4392 = 160933 = 185230 Reciprocals. 39370 032809 010936 6-2138 x 10- 6 5'398xlO- 6 More exactly, according to Captain Clarke's compari- sons of standards of length (printed in 1866), the metre is equal to 1-09362311 yard, or 3 -2808693 feet, or 39'370432 inches, the standard metre being taken as correct at C., and the standard yard as correct at 16f C. Hence the inch is 2-5399772 centimetres. Area. 1 sq. inch, - 1 sq. foot, - 1 sq. yard, - 1 sq. mile, - sq. cm. - = 6-4516 - = 929-01 - =8361-13 - = 2-59 x 10 10 A Reciprocals. 1550 001076 0001196 3-861 x 10-" 2 UNITS AND PHYSICAL CONSTANTS. Volume. cub. cm. Reciprocals. 1 cubic inch, - = 16 '387 '06102 1 cubic foot, - -28316- 3'532xlO- 5 1 cubic yard, = 764535' 1 "308 x 10~ 6 Ipint,- - = 567-63 '001762 1 gallon, - = 4541' '0002202 Mass. gm. Reciprocals. 1 grain, - '0647990 15 '432 1 ounce avoir., - = 28 '3495 -035274 1 pound - =453-59 '0022046 1 ton, - - - = 1-01605 xlO 6 9'84206xlO- 7 According to the comparison made by Professor W. H. Miller in 1844 of the "kilogramme des Archives," the standard of French weights, with two English pounds of platinum, and additional weights, also of platinum, the kilo- gramme is 1 5432*34874 grains, of which the new standard pound contains 7000. Hence the kilogramme would be 2-2046212 pounds, and the pound 453*59265 grammes. Three standard pounds, one of platinum-indium and the other two of gilded bronze, belonging to the Standards Department, were compared, in 1883, at the Bureau In- ternational des Poids et Mesures, with standards belong- ing to the Bureau, and their values in grammes were found to be respectively 453-59135, 453-58924, 453-58738. Travaux et Memoir es, tome IV. Velocity. cm. per sec. Reciprocals. 1 foot per second, - - =30 "4797 '032809 1 statute mile per hour, - =44 '704 -022369 1 nautical mile per hour, - =51 -453 '019435 1 kilometre per hour, - =27 '777 '036 TABLES. Acceleration. cm. per sec. per sec. Reciprocal. 1 ft. per sec. per sec., - =30 '4797 "032809 Density. gm. per c.c. Reciprocals. 1 lb. per cubic foot, ' - - = "016019 62'426 1 grain per cubic inch, - = '003954 252 '88 Stress (in gravitation measure}. gm. per sq. cm. Reciprocals. 1 lb. per sq. foot, - = "48826 2 "0481 1 lb. per sq. inch, - = 70'31 '014223 1 inch of mercury at j = 34>534 .^^ 30 inches ,, ,, = 1036"0 '00096525 760mm. ,, =1033-3 "00096777 Surface Tension (in gravitation measure). gm. per cm. Reciprocals. 1 grain per linear inch, - "02551 39 "20 lib. foot, - =14-88 "06720 Work (in gravitation measure). gm.-cm. Reciprocals. 1 foot-pound, - =13825 7"2331xlO~ 5 1 foot-grain, - = 1 -975 "50632 1 foot-ton, - - = 3'097xl0 7 6 "494 x 10 - & 1 kilogram metre, - = 10 5 10 ~ 5 Rate of Working (in gravitation measure). gm.-cm. per sec. Reciprocals. 1 horse-power, - = 7'604xl0 6 l"3151x!0- 7 1 force-de-cheval, - = 7'SxlO 6 l'3333xlO- 7 Heat (in gravitation measure). gm.-cm. Reciprocals. 1 gm. deg., - - =42400 2'36xlO- 5 1 lb. deg. Cent., - = l'923x!0 7 5"2xlO- 8 1 ,, Fahr., - = l"068x!07 9"36xlQ- 8 The following reductions of gravitation measures to absolute measures are on the assumption that # = 981 : UNITS AND PHYSICAL CONSTANTS. Force (in absolute measure). Dynes. Reciprocals. Weight of 1 gm., - - = 981 '001019 Ikilogm., - = 9-81 xlO 5 1 '019x10-* 1 tonne, - = 9'81 x 10* l'019x!0- 9 Iton, . - = 9-97 xlO 8 1'003 x lO' 9 Icwt.,- - = 4-98 xlO 7 2-008x10-* 1 Ib. avoir., - = 4'45 x 10 5 2'247 x 10-" 1 oz. - = 2'78^< 10 4 3-597 x 10- 5 1 grain, - = 63'57 '01573 i puandal, =13825 7-2333x10-- (The ratio of the poundal to the dyne is independent of g. ) Stress (in absolute measure). Dynes per sq. cm. Reciprocals 1 Ib. per sq. foot, - = '479 '00209 lib. inch, - = 6'9 xlO 4 1 -45xlO- 5 1 gm. cm., '981 '00102 1 kilo. decim., - = 9'SlxlO 3 l'02xlO- 4 1 cm. of mercury at C. , = 13338' '0000736 76 = 1-0136 xlO 6 9-866 xlO- 7 1 inch = 3-388 x 10 4 2'95 x 10~ 5 30 l-0163x!0 6 9-84xlO- 7 Surface Tension (in absolute measure}. Dynes per cm. Reciprocals. Igm. per linear cm., = 981 '00102 1 grain ,, inch, - = 25 '04 lib. foot, - - =14600 6-85 x 10-* Work and Energy (in absolute measure). Ergs. Reciprocals. 1 gm. cm., - - = 981 '001019 1 kilogrammetre, = 9 '81 x 10 7 1 '019 x 10~ 8 1 foot-pound, - = 1-356x107 7 '37x10-* 1 foot-poundal, - =421390 2-3731 x 10~ 6 (The ratio of the ft. -poundal to the erg is independent of g.) ] joule - - = 10 7 ergs. Rate of Working (in absolute measure). Ergs per sec. Reciprocals. 1 horse-power, - - = 7 '46 x 1 9 1 '34 x 1 - 10 1 force-de-cheval, - - =7'36xl0 9 l'36xlO- 1( > 1 watt, - - - - = 10 7 lO- 7 Heat (in absolute measure). Ergs. Reciprocals. Igm. deg., - - - =4-2 xlO 7 2'38xlO- 8 ! deg. Cent., - - =l'905x!0 10 S^SxlQ- 11 Fahr., - - =l'058x 10 10 9-45xlO' n 1 gm I Ib. CHAPTER I. GENERAL THEORY OF UNITS. Units and Derived Units. 1. THE numerical value of a concrete quantity is its ratio to a selected magnitude of the same kind, called the unit. Thus, if L denote a definite length, and I the unit length, - is a ratio in the strict Euclidian sense, and is called the numerical value of L. The numerical value of a concrete quantity varies directly as the concrete quantity itself, and inversely as the unit in terms of which it is expressed. 2. A unit of one kind of quantity is sometimes defined by reference to a unit of another kind of quantity. For example, the unit of area is commonly defined to be the area of the square described upon the unit of length ; and the unit of volume is commonly defined as the volume of the cube constructed on the unit of length. The units of area and volume thus defined are called derived unite, and are more convenient for calculation than indepen- dent units would be. For example, when the above 6 UNITS AND PHYSICAL CONSTANTS. [CHAP. definition of the unit of area is employed, we can assert that [the numerical value of] the area of any rectangle is equal to the product of [the numerical values of] its length and breadth ; whereas, if any other unit of area were employed, we should have to introduce a third factor which would be constant for all rectangles. 3. Still more frequently, a unit of one kind of quantity is defined by reference to two or more units of other kinds. For example, the unit of velocity is commonly defined to be that velocity with which the unit length would be described in the unit time. When we specify a velocity as so many miles per hour, or so many feet per second, we in effect employ as the unit of velocity a mile per hour in the former case, and a foot per second in the latter. These are derived units of velocity. Again, the unit acceleration is commonly defined to be that acceleration with which a unit of velocity would be gained in a unit of time. The unit of acceleration is thus derived directly from the units of velocity and time, and therefore indirectly from the units of length and time. 4. In these and all other cases, the practical advantage of employing derived units is, that we thus avoid the intro- duction of additional factors, which would involve needless labour in calculating and difficulty in remembering. * 5. The correlative term to derived is fundamental. Thus, when the units of area, volume, velocity, and * An example of such needless factors may be found in the rules commonly given in English books for finding the mass of a body when its volume and material are given. ' ' Multiply the volume in cubic feet by the specific gravity and by 62 '4, and the product will be the mass in pounds ; " or " multiply the volume in cubic i.] GENERAL THEORY OF UNITS. 7 acceleration are defined as above, the units of length and time are called the fundamental units. Dimensions. 6. Let us now examine the laws according to which derived units vary when the fundamental units are changed. Let V denote a concrete velocity such that a concrete length L is described in a concrete time T ; and let v, I, t denote respectively the unit velocity, the unit length, and the unit time. The numerical value of V is to be equal to the numerical value of L divided by the numerical value of T. But V L T these numerical values are , , : v L t hence we must have Hi- o This equation shows that, when the units are changed (a change which does not affect V, L, and T), v must vary directly as I and inversely as t ; that is to say, the unit of velocity varies directly as the unit of length, and inversely as the unit of time. V Equation (1) also shows that the numerical value of a given velocity varies inversely as the unit of length, and directly as the unit of time. inches by the specific gravity and by 253, and the product will be the mass in grains." The factors 62 '4 and 253 here employed would be avoided that is, would be replaced by unity, if the unit volume of water were made the unit of mass. 8 UNITS AND PHYSICAL CONSTANTS. [CHAP. 7. Again, let A denote a concrete acceleration such that the velocity Y is gained in the time T", and let a denote the unit of acceleration. Then, since the numerical value of the acceleration A is the numerical value of the velocity Y divided by the numerical value of the time T', we have But by equation (1) we may write -for. We I T v thus obtain = a I T T' This equation shows that when the units , I, t are changed (a change which will not affect A, L, T or T'), a must vary directly as Z, and inversely in the duplicate ^ ratio of t ; and the numerical value will vary inversely a as I, and directly in the duplicate ratio of t. In other words, the unit of acceleration varies directly as the unit of length, and inversely as the square of the unit of time; and the numerical value of a given acceleration varies inversely as the unit of length, and directly as the square of the unit of time. It will be observed that these have been deduced as direct consequences from the fact that [the numerical value of] an acceleration is equal to [the numerical value of] a length, divided by [the numerical value of] a time, and then again by [the numerical value of] a time. The relations here pointed out are usually expressed by i.] GENERAL THEORY OF UNITS. 9 savins: that the dimensions of acceleration* are ,- , or (time)* that the dimensions of the unit of acceleration* are unit of length (unit of time) 2 8. We have treated these two cases very fully, by way of laying a firm foundation for much that is to follow. We shall hereafter use an abridged form of reasoning, such as the following : , .. length velocity = . - : time , ,. velocity length acceleration = : - = T - r ^~--. time Such equations as these may be called dimensional equations. Their full interpretation is obvious from what precedes. In all such equations, constant numerical factors can be discarded, as not affecting dimensions. 9. As an example of the application of equation (2) we shall compare the unit acceleration based on the foot and second with the unit acceleration based on the yard and minute. Let I denote a foot, L a yard, t a second, T a minute, T' a minute. Then a will denote the unit acceleration based on the foot and second, and A will denote the unit * Professor James Thomson ('Brit. Assoc. Report,' 1878, p. 452) objects to these expressions, and proposes to substitute the following : "Change-ratio of ut.it of acceleration =^^ggS&" This is very clear and satisfactory as a full statement of the meaning intended ; but it is necessary to tolerate some abridg- ment of it for practical working. 10 UNITS AND PHYSICAL CONSTANTS. [CHAP, acceleration based on the yard and minute. Equation (2) becomes A_3 /1\ 2 _ 1 ,ov ^~I X V60J~1200 ; that is to say, an acceleration in which a yard per minute of velocity is gained per minute, is *- of an acceleration 1200 in which a foot per second is gained per second. Meaning of "per." 10. The word per, which we have several times em- ployed in the present chapter, denotes division of the quantity named before it by the quantity named after it. Thus, to compute velocity in feet per second, we must divide a number of feet by a number of seconds.* If velocity is continuously varying, let x be the number of feet described since a given epoch, and t the number of seconds elapsed, then - - is what is meant by the at number of feet per second. The word should never be employed in the specification of quantities, except when the quantity named before it varies directly as the quantity named after it, at least for small variations as, in the above instance, the distance described is ultimately pro- portional to the time of describing it. Extended Sense of the terms " Multiplication " and 11 Division" 11. In ordinary multiplication the multiplier is always- * It is not correct to speak of interest at the rate of Five Pounds per cent. It should be simply Five pzr cent. A rate of five pounds in every hundred pounds is not different from a rate of five shillings in every hundred shillings. i.] GENERAL THEORY OF UNITS. 11 a mere numerical quantity, and the product is of the same nature as the multiplicand. Hence in ordinary division either the divisor is a mere numerical quantity and the quotient a quantity of the same nature as the dividend ; or else the divisor is of the same nature as the dividend, and the quotient a mere numerical quantity. But in discussing problems relating to units, it is con- venient to extend the meanings of the terms "multiplica- tion " and " division." A distance divided by a time will denote a velocity the velocity with which the given distance would be described in the given time. The dis- tance can be expressed as a unit distance multiplied by a numerical quantity, and varies jointly as these two factors : the time can be expressed as a unit timo multiplied by a numerical quantity, and is jointly proportional to these two- factors. Also, the velocity remains unchanged when the time and distance are both changed in the same ratio. 12. The three quotients 1 mile 5280 ft. 22ft. 1 hour' 3600 sec.' 15 sec. all denote the same velocity, and are therefore to be regarded as equal. In passing from the first to the second, we have changed the units in the inverse ratio to their numerical multipliers, and have thus left both the distance and the time unchanged. In passing from the second to the third, we have divided the two numeri- cal factors by a common measure, and have thus changed 'the distance and the time in the same ratio. A change in either factor of the numerator will be compensated by a proportional change in either factor of the denom- inator. 1 2 UNITS AND PHYSICAL CONSTANTS. [CHAP. Further, since the velocity -' - is -- of the velo- 15 sec. 15 .. 1 ft. 22 ft. 22 ft. -city v . we are entitled to write - - . , 1 sec. 15 sec. 15 sec. thus separating the numerical part of the expression from the units part. In like manner we may express the result of Art. 9 by writing yard 1 foot (minute) 2 1200 ' (second) 2 Such equations as these may be called " physical equations," inasmuch as they express the equality of physical quantities, whereas ordinary equations express the equality of mere numerical values. The use of physical equations in problems relating to units is to be strongly recommended, as affording a natural and easy clue to the necessary calculations, and especially as obviating the doubt by which the student is often embarrassed as to whether he ought to multiply or divide. 13. In the following examples, which illustrate the use of physical equations, we shall employ I to denote the unit length, m the unit mass, and t the unit time. Ex. 1. If the yard be the unit of length, and the acceleration of gravity (in which a velocity of 32*2 ft. per sec. is gained per sec.) be represented by 2415, find the unit of time. We have I = yard, and 32-2 -A = 2415 l - = 2415 ~ (sec.) 2 t* t 2 t' 2 = p t sec.* = 225 sec. 2 , t = 15 sec. i.] GENERAL THEORY OF UNITS. 13 Ex. 2. If the unit time be the second, the unit density 162 Ibs. per cub. ft., and the unit force * the weight of an ounce at a place where g (in foot-second units) is 32 r what is the unit length 1 We have , = sec., .- and - 32 . , or ml =32 oz. ft. = 2 Ib. ft. sec. 2 sec.- Hence by division I* - ^ (ft.)*, jr.'. J It - 4 in. Ex. 3. If the area of a field of 10 acres be represented by 100, and the acceleration of gravity (taken as 32 foot- second units) be 58 1, find the unit of time. We have 48400 (yd.) 2 -100 l\ whence 1 = 22 yd.; and , (sec.) 2 ' ? 3 t- whence tf = ^- sec. 2 = 121 sec. 2 , =11 sec. Ex. 4. If 8 ft. per sec. be the unit velocity, and the acceleration of gravity (32 foot-second units) the unit acceleration, find the units of length and time. We have the two equations I -, ft. I oo ft- = o - , = o^ - , t sec. t- sec.- whence by division t = \ sec., and substituting this value of t in the first equation, we have 4 =8 ft., 1=2 ft. Ex. 5. If the unit force be 100 Ibs. weight, the unit length 2 ft., and the unit time J sec., find the unit mass, the acceleration of gravity being taken as 32 foot-second units. * For the dimensions of density and force, see Art. 14. 14 UNITS AND PHYSICAL CONSTANTS. [CHAP. We have I = 2 ft., t = J sec., 100 Ib. 32 J?L- = ^L = ^_2J*r sec. 2 tf Jg- sec. 2 that is 100 x 32 Ib. = 32 m, m = 100 Ib. Ex. 6. The number of seconds in the unit of time is equal to the number of feet in the unit of length, the unit of force is 750 Ibs. weight [g being 32], and a cubic foot of the standard substance [substance of unit density] con- tains 13500 oz. Find the unit of time. Let t = x sec., then l = xft. also let m = ylb. Then we have ml = y Ib. x ft. _ y Ib. ft. = 75Q x g9 lb. ft. tf x 2 sec. 2 x sec. 2 sec. 2 or = 750 x 32. x AT m. y Ib. IOCAA Z - Also T -I |p- 13500 a3 ; whence U - 13500 x I. x 6 lo Hence by division 750 x 32 x 16 16 2 16 , 16 ,. Ex. 7. When an inch is the unit of length and t the unit of time, the measure of a certain acceleration is a ; when 5 ft. and 1 min. are the units of length and time respectively, the measure of the same acceleration is 10 a. Find t. Equating the two expressions for the acceleration, we , inch , ~ 5 ft. have a = 10 a - : - , t~ (mm.) 2 i.] GENERAL THEORY OF UNITS. 15 ,.0 / \o inch (min.) 2 n / >> whence t' = (mm.)" = = 6 (sec.)" 7 50 ft. 600 t ^6 sec. Ex. 8. The numerical value of a certain force is 56 when the pound is the unit of mass, the foot the unit of length, and the second the unit of time ; what will be the numerical value of the same force when the hundredweight is the unit of mass, the yard the unit of length, and the minute the unit of time 1 Denoting the required value by x we have sec." mm.- Ib. ft. x = 56 cwt ft. /min.V : ^d. \ sea / = 56 x _L x 1 x 60 2 = 600. Dimensions of Mechanical and Geometrical Quantities. 14. In the following list of dimensions, we employ the letters L, M, T as abbreviations for the words Length, Mass, Time. The symbol of equality is used to denote sameness of dimensions. Area = L 2 , Volume - L 3 , Velocity = -, Acceleration = , Momentum = -~. Density = , density being defined as mass per unit volume. Force = -, since a force is measured by the momen- tum which it generates per unit of time, and is therefore 16 UNITS AND PHYSICAL CONSTANTS. [CHAP, the quotient of momentum by time or since a force is- measured by the product of a mass by the acceleration generated in this mass. Work = , being the product of force and distance. Kinetic energy = -=^-, being half the product of mass by square of velocity. The constant factor J can be omitted, as not affecting dimensions. ML 2 Moment of couple = ^ , being the product of a force by a length. The dimensions of angle* when measured by arc radius are zero. The same angle will be denoted by the same number, whatever be the unit of length employed. In /. . i arc L T ft fact we have - = - == L. radius L The work done by a couple in turning a body through any angle, is the product of the couple by the angle. The identity of dimensions between work and couple is thus verified. Angular velocity Angular acceleration = . Moment of inertia = ML 2 . ML 2 Angular momentum = moment of momentum = , * The name radian has been given to the angle whose arc is equal to radius. "An angle whose value in circular measure is 6 " is " an angle of radians." i.] GENERAL THEORY OF UNITS. 17 being the product of moment of inertia by angular velo- city, or the product of momentum by length. Intensity of pressure, or intensity of stress generally, being force per unit of area, is of dimensions - : that area Intensity of force of attraction at a point, often called simply force at a point, being force per unit of attracted mass, is of dimensions - - or . It is numerically mass T 2 equal to the acceleration which it generates, and has accordingly the dimensions of acceleration. The absolute force of a centre of attraction, better called the strength of a centre, may be defined as the intensity of force at unit distance. If the law of attraction be that of inverse squares, the strength will be the product of the intensity of force at any distance by the square of this L 3 distance, and its dimensions will be . Curvature (of a curve) = , being the angle turned by -L the tangent per unit distance travelled along the curve. Tortuosity = , being the angle turned by the osculat- Ju ing plane per unit distance travelled along the curve. The solid angle or aperture of a conical surface of any form is measured by the area cut off by the cone from a sphere whose centre is at the vertex of the cone, divided by the square of the radius of the sphere. Its dimensions are therefore zero ; or a solid angle is a numerical quan- tity independent of the fundamental units. 18 UNITS AND PHYSICAL CONSTANTS. [CHAP. i. The specific curvature of a surface at a given point (Gauss's measure of curvature) is the solid angle de- scribed by a line drawn from a fixed point parallel to the normal at a point which travels on the surface round the given point, and close to it, divided by the very small area thus enclosed. Its dimensions are therefore . The mean curvature of a surface at a given point, in the theory of Capillarity, is the arithmetical mean of the curvatures of any two normal sections normal to each other. Its dimensions are therefore . Ju 19 CHAPTER II. CHOICE OF THREE FUNDAMENTAL UNITS. 15. NEARLY all the quantities with which physical science deals can be expressed in terms of three funda- mental units ; and the quantities commonly selected to serve as the fundamental units are a definite length, a definite mass, a definite interval of time. This particular selection is a matter of convenience rather than of necessity ; for any three independent units are theoretically sufficient. For example, we might em- ploy as the fundamental units a definite mass, a definite amount of energy, a definite density. 16. The following are the most important considera- tions which ought to guide the selection of fundamental units : (1) They should be quantities admitting of very accurate comparison with other quantities of the same kind. 20 UNITS AND PHYSICAL CONSTANTS. [CHAP. (2) Such comparison should be possible at all times. Hence the standards must be permanent that is, not liable to alter their magnitude with lapse of time. (3) Such comparisons should be possible at all places. Hence the standards must not be of such a nature as to change their magnitude when carried from place to place. (4) The comparison should be easy and direct. Besides these experimental requirements, it is also desirable that the fundamental units be so chosen that the definition of the various derived units shall be easy, and their dimensions simple. 17. There is probably no kind of magnitude which so completely fulfils the four conditions above stated as a standard of mass, consisting of a piece of gold, platinum, or some other substance not liable to be affected by atmospheric influences. The comparison of such a standard with other bodies of approximately equal mass is effected by weighing, which is, of all the operations of the laboratory, the most exact. Very ac- curate copies of the standard can thus be secured; and these can be carried from place to place with little risk of injury. The third of the requirements above specified forbids the selection of a force as one of the fundamental units. Forces at the same place can be very accurately measured by comparison with weights; but as gravity varies from place to place, the force of gravity upon a piece of metal, or other standard weight, changes its magnitude in travelling from one place to another. A spring-balance, it is true, gives a direct measure of IT.] THREE FUNDAMENTAL UNITS. 21 force ; but its indications are too rough for purposes of accuracy. 18. Length is an element which can be very accurately measured and copied. But every measuring instrument is liable to change its length with temperature. It is therefore necessary, in denning a length by reference to a concrete material standard, such as a bar of metal, to state the temperature at which the standard is correct. The temperature now usually selected for this purpose is that of a mixture of ice and water (0 C.), observation having shown that the temperature of such a mixture is constant. The length of the standard should not exceed the length of a convenient measuring-instrument ; for, in comparing the standard with a copy, the shifting of the measuring- instrument used in the comparison introduces additional risk of error. In end-standards, the standard length is that of the bar as a whole, and the ends are touched by the instrument every time that a comparison is made. This process is liable to wear away the ends and make the standard false. In line-standards, the standard length is the distance be- tween two scratches, and the comparison is made by optical means. The scratches are usually at the bottom of holes sunk halfway through the bar. 19. Time is also an element which can be measured with extreme precision. The direct instruments of mea- surement are clocks and chronometers ; but these are checked by astronomical observations, and especially by transits of stars. The time between two successive tran- sits of a star is (very approximately) the time of the 22 UNITS AND PHYSICAL CONSTANTS. [CHAP. earth's rotation on its axis ; and it is upon the uniformity of this rotation that the preservation of our standards of time depends. Necessity for a Common Scale. 20. The existence of quantitative correlations between the various forms of energy, imposes upon men of science the duty of bringing all kinds of physical quantity to one common scale of comparison. Several such measures (called absolute measures) have been published in recent years ; and a comparison of them brings very promi- nently into notice the great diversity at present existing in the selection of particular units of length, mass, and time. Sometimes the units employed have been the foot, the grain, and the second ; sometimes the millimetre, milli- gramme, and second ; sometimes the centimetre, gramme, and second ; sometimes the centimetre, gramme, and minute ; sometimes the metre, tonne, and second ; some- times the metre, gramme, and second ; while sometimes a mixture of units has been employed ; the area of a plate, for example, being expressed in square metres, and its thickness in millimetres. A diversity of scales may be tolerable, though undesir- able, in the specification of such simple matters as length, area, volume, and mass when occurring singly ; for the reduction of these from one scale to another is generally understood. But when the quantities specified involve a reference to more than one of the fundamental units,, and especially when their dimensions in terms of these units are not obvious, but require careful working out, ii.] THREE FUNDAMENTAL UNITS. 23 there is great increase of difficulty and of liability to mistake. A general agreement as to the particular units of length, mass, and time which shall be employed if not in all scientific work, at least in all work involving complicated references to units is urgently needed ; and almost any one of the selections above instanced would be better than the present option. 21. We shall adopt the recommendation of the Units Committee of the British Association (see Appendix), that all specifications be referred to the Centimetre, the Gramme, and the Second. The system of units derived from these as the fundamental units is called the C.G.S. system; and the units of the system are called the C.G.S. units. The reason for selecting the centimetre and gramme, rather than the metre and gramme, is that, since a gramme of water has a volume of approximately 1 cubic centimetre, the former selection makes the density of water unity; whereas the latter selection would make it a million, and the density of a substance would be a million times its specific gravity, instead of being identical with its specific gravity as in the C.G.S. system. Even those who may have a preference for some other units will nevertheless admit the advantage of having a variety of results, from various branches of physics, re- duced from their original multiplicity and presented in one common scale. 22. The adoption of one common scale for all quan- tities involves the frequent use of very large and very 24 UNITS AND PHYSICAL CONSTANTS. [CHAP. n. small numbers. Such numbers are most conveniently written by expressing them as the product of two factors, one of which is a power of 10 ; and it is usually advan- tageous to effect the resolution in such a way that the exponent of the power of 10 shall be the characteristic of the logarithm of the number. Thus 3240000000 will be written 3-24 x 10 9 , and '00000324 will be written 3-24 x 10- 6 . 25 CHAPTER III. MECHANICAL UNITS. Value of g. 23. ACCELERATION is defined as the rate of increase of velocity per unit of time. The C.G.S. unit of accelera- tion is the acceleration of a body whose velocity increases in every second by the C.G.S. unit of velocity namely, by a centimetre per second. The apparent acceleration of a body falling freely under the action of gravity in vacuo is denoted by g. The value of g in C.G.S. units at any part of the earth's surface is approximately given by the following formula, g = 980-6056 - 2-5028 cos 2/X - -000003/i, A denoting the latitude, and h the height of the station (in centimetres) above sea-level. The constants in this formula have been deduced from numerous pendulum experiments in different localities, the length I of the seconds' pendulum being connected with the value of g by the formula g = Tr 2 l. Dividing the above equation by ?r 2 we have, for the length of the seconds' pendulum, in centimetres, I = 99-3562 - -2536 cos 2A - -0000003A. 26 UNITS AND PHYSICAL CONSTANTS. [CHAP. At sea-level these formulae give the following values for the places specified : Latitude. Value of g. Value of I. Equator, - 978-10 99-103 Latitude 45, 45 980-61 99-356 Munich, - 48 9 980-88 99-384 Paris, 48 50 980-94 99-390 Greenwich, 51 29 981-17 99-413 Gottingen, 51 32 981-17 99-414 Berlin, - 52 30 981-25 99-422 Dublin, - 53 21 981-32 99-429 Manchester, 53 29 981-34 99-430 Belfast, - 54 36 981-43 99-440 Edinburgh, 55 57 981-54 99-451 Aberdeen, 57 9 981-64 99-461 Pole, 90 983-11 99-610 The difference between the greatest and least values (in the case of both g and 1) is about of the mean j. y o value. 24. The Standards Department of the Board of Trade, being concerned only with relative determinations, has adopted the formula g = <7 (1- -00257 cos 2A)(l - | A), A. denoting the latitude, k the height above sea-level, K the earth's radius, g the value of g in latitude 45 at sea- level, which may be treated as an unknown constant multiplier. Putting for E, its value in centimetres, the formula gives g = g (l- -00257 cos 2A.-1 -96/i x 10' 9 ), where h denotes the height in centimetres. in.] MECHANICAL UNITS. 27 The formula which we employed in the preceding section gives As regards the factor dependent on height, theory indi- cates 1 - as its correct value for such a case as that of XV a balloon in mid-air over a low-lying country ; the value 5 h 1 -- may be accepted as more correct for an elevated 4: .TV plateau on the earth's surface. Force. 25. The C.G.S. unit of force is called the dyne. It is the force which, acting upon a gramme for a second, generates a velocity of a centimetre per second. It may otherwise be denned as the force which, acting upon a gramme, produces the C.G.S. unit of acceleration, or as the force which, acting upon any mass for 1 second, produces the C.G.S. unit of momentum. To show the equivalence of these three definitions, let m denote mass in grammes, v velocity in centimetres per second, t time in seconds, F force in dynes. Then, by the second law of motion, we have , . . force acceleration = - : mass "IT that is, if a denote acceleration in C.G.S. units, a=- ; m hence, when a and m are each unity, F will be unity. Again, by the nature of uniform acceleration, we have v = at, v denoting the velocity due to the acceleration a, continuing for time t. 28 UNITS AND PHYSICAL CONSTANTS. [CHAP. Hence we have F = ma = . Therefore, if mv = 1 and t=l t we have F = 1. As a particular case, if w = l, vl, t = l, we have 26. The force represented by the weight of a gramme varies from place to place. It is the force required to sustain a gramme in vacuo, and would be nil at the earth's centre, where gravity is nil. To compute its amount in dynes at any place where g is known, observe that a mass of 1 gramme falls in vacuo with acceleration g. The force producing this acceleration (namely, the weight of the gramme) must be equal to the product of the mass and acceleration, that is, to g. The weight (when weight means force) of 1 gramme is therefore g dynes ; and the weight of in grammes is mg dynes. 27. Force is said to be expressed in gravitation-measure when it is expressed as equal to the weight of a given mass. Such specification is inexact unless the value of g is also given. For purposes of accuracy it must always be remembered that the pound, the gramme, etc., are, strictly speaking, units of mass. Such an expression as " a force of 100 tons " must be understood as an abbrevia- tion for " a force equal to the weight [at the locality in question] of 100 tons." 28. The name poundal has recently been given to the unit force based on the pound, foot, and second ; that is, the force which, acting on a pound for a second, gene- rates a velocity of a foot per second. It is of the in.] MECHANICAL UNITS. 29 weight of a pound, y denoting the acceleration due to gravity expressed in foot-second units, which is about 32-2 in Great Britain. To compare the poundal with the dyne, let x denote the number of dynes in a poundal ; then we have mn. cm. Ib. ft. x 1 - sec." 1 sec.- x = . = 453-59 x 30-4797 = 13825. gm. cin. Work and Energy. 29. The C.G.S. unit of work is called the erg. It is the amount of work done by a dyne working through a distance of a centimetre. The C.G.S. unit of energy is also the erg, energy being measured by the amount of work which it represents. 30. To establish a rule for computing the kinetic energy (or energy due to the motion) of a given mass moving with a given velocity, it is sufficient to consider the case of a body falling in vacuo. When a body of m grammes falls through a height of h centimetres, the working force is the weight of the body that is, gm dynes, which, multiplied by the distance worked through, gives gmh ergs as the work done. But the velocity acquired is such that v~ = 2gh. Hence we have gmh = ^mv 2 . The kinetic energy of a mass of m grammes moving with a velocity of v centimetres per second is therefore ^mv 2 ergs ; that is to say, this is the amount of work which would be required to generate the motion of the body, or is the amount of work which the body 30 UNITS AND PHYSICAL CONSTANTS. [CHAP. would do against opposing forces before it would coine to rest. 31. Work, like force, is often expressed in gravitation- measure. Gravitation units of work, such as the foot- pound and kilogramme-metre, vary with locality, being proportional to the value of g. One gramme-centimetre is equal to g ergs. One kilogramme-metre is equal to 100,000 g ergs. One foot-poundal is 453'59 x (30'4797) 2 = 421390 ergs. One foot-pound is 13,825 g ergs, which, if g be taken as 981, is 1-356 x 10 7 ergs. 32. The C.G.S. unit rate of working is 1 erg per second. Watt's " horse-power " is denned as 550 foot-pounds per second. This is 7 '46 x 10 9 ergs per second. The " force de cheval " is denned as 75 kilogrammetres per second. This is 7 '36 x 10 9 ergs per second. We here assume g 981. A new unit of rate of working has been lately intro- duced for convenience in certain electrical calculations. It is called the Watt, and is denned as 10 7 ergs per second. A thousand watts make a kilowatt. The following tabular statement will be useful for reference. 1 Watt = 10 7 ergs per second = '00134 horse-power = 737 foot-pounds per second = '101 9 kilogram- metres per second. 1 Kilowatt = 1'34 horse-power. 1 Horse-power = 550 foot-pounds per second = 76'0 kilogrammetres per second = 746 watts = 1 '01385 force de cheval. 1 Force de cheval = 75 kilogrammetres per second = 542 '48 foot-pounds per second = 736 watts = '9863 horse-power. m.l MECHANICAL UNITS. 31 Examples. 1. If a spring balance is graduated so as to show the masses of bodies in pounds or grammes when used at the equator, what will be its error when used at the poles, neglecting effects of temperature 1 Ans. Its indications will be too high by about j-gg of the total weight. 2. A cannon-ball, of 10,000 grammes, is discharged with a velocity of 45,000 centims. per second. Find its kinetic energy. Ans. | x 10000 x (45000) 2 = 1-0125 x 10 13 ergs. 3. In last question find the mean force exerted upon the ball by the powder, the length of the barrel being 200 centims. Ans. 5-0625 x 10 10 dynes. 4. Given that 42 million ergs are equivalent to 1 gramme-degree of heat, and that a gramme of lead at 10 C. requires 15 '6 gramme-degrees of heat to melt it; find the velocity with which a leaden bullet must strike a target that it may just be melted by the collision, suppos- ing all the mechanical energy of the motion to be converted into heat and to be taken up by the bullet. We have Jv 2 = 15-6 x J, where J = 42 x 10 6 . Hence i> 2 =1310 millions; -v-36'2 thousand centims. per second. 5. With what velocity must a stone be thrown verti- cally upwards at a place where gis 981 that it may rise to a height of 3000 centims. ? and to what height would it ascend if projected vertically with this velocity at the surface of the moon, where g is 150 1 Ans. 2426 centims. per second ; 19620 centims. 32 UNITS AND PHYSICAL CONSTANTS. [CHAP. Centrifugal Force. 33. A body moving in a curve must be regarded as continually falling away from a tangent. The accelera- tion with which it falls away is , v denoting its velocity and r the radius of curvature. The acceleration of a body in any direction is always due to force urging it in that direction, this force being equal to the product of mass and acceleration. Hence the normal force on a body of in grammes moving in a curve of radius r centimetres, with velocity v centimetres per second, is dynes. This force is directed towards the centre of curvature. The equal and opposite force with which the body reacts is called centrifugal force. If the body moves uniformly in a circle, the time of revolution being T seconds, we have v = ^ ', hence ; == (^p~) r > an d * ne force acting on the body is nes. If n revolutions are made per minute, the value of T is 60 (mr\- , , and the force is mr\ ^1 dynes. Examples. 1. A body of m grammes moves uniformly in a circle of radius 80 centims., the time of revolution being J of a in.] MECHANICAL UNITS. 33 second. Find the centrifugal force, and compare it with the weight of the body. Ans. The centrifugal force ismx/-^j x80 = mx 647r 2 x 80 = 50532 m dynes. The weight of the body (at a place where g is 981) is 981 m- dynes. Hence the centrifugal force is about 52| times the weight of the body. .2. At a bend in a river, the velocity in a certain part of the surface is 170 centims. per second, and the radius of curvature of the lines of flow is 9100 centims. Find the slope of the surface in a section transverse to the lines of flow. Ans. Here the centrifugal force for a gramme of the water is ( 17 ^ 8 = 3-176 dynes. If g be 98 1 the slope will o .1 fr f* 1 be = ; that is, the surface will slope upwards i/ol oUJ from the concave side at a gradient of 1 in 309. The general rule applicable to questions of this kind is that the resultant of centrifugal force and gravity must be normal to the surface. 3. An open vessel of liquid is made to rotate rapidly round a vertical axis. Find the number of revolutions that must be made per minute in order to obtain a slope of 30 at a part of the surface distant 10 centims. from the axis, the value of g being 981. Ans. We must have tan 30= /, where /denotes the 9 intensity of centrifugal force that is, the centrifugal force per unit mass. We have therefore c 34 UNITS AND PHYSICAL CONSTANTS. [CHAP. 981 tan 30 = lof Y' n denotin the number of \30/ revolutions per minute, w V ~~90~* Hence rc = 71'9. 4. For the intensity of centrifugal force at the equator due to the earth's rotation, we have r = earth's radius = 6-38 x 10 8 , T = 86164, being the number of seconds in a sidereal day. This is about of the value of g. If the earth were at rest, the value of g at the equator would be greater than at present by this amount. If the earth were revolving about 17 times as fast as at present, the value of g at the equator would be nil. SUPPLEMENTAL SECTION. On the help to be derived from Dimensions in investi- gating Physical Formulae. -. When one physical quantity is known to vary as some power of another physical quantity, it is often possible to find the exponent of this power by reasoning based on dimensions, and thus to anticipate the results or some of the results of a dynamical investigation. Examples. 1. The time of vibration of a simple pendulum in a small arc depends on the length of the pendulum and the intensity of gravity. If we assume it to vary as the m th in.] MECHANICAL UNITS. 35 power of the length, and as the n th power of #, and to be independent of everything else, the dimensions of a time must equal the m th power of a length, multiplied by the n th power of an acceleration, that is T = LLT- 2M = L w L M T- 2n Since the dimensions of both members are to be identical, we have, by equating the exponents of T, 1 = - 2n, whence n = - J, and by equating the exponents of L, m + n = 0, whence m = J ; that is, the time of vibration varies directly as the square root of the length, and inversely as the square root of g. 2. The velocity of sound in a gas depends only on the density D of the gas and its coefficient of elasticity E, and we shall assume it to vary as D m E". The dimensions of velocity are LT~ T . The dimensions of density, or - - , are ML~ 3 . volume The dimensions of E, which will be explained in the chapter on stress and strain, are - , or (MLT' 2 )L~ 2 , or area ML- 1 T- 2 . The equation of dimensions is LT _i = MW L _ 3m MW L _ n T _ 2M) = M m+ " L~ 3w - n T~ 2w , whence, by equating coefficients, we have the three equations 1 = 3m ri) 1 = 2n, m + n = 0, to determine the two unknowns m and n. 36 UNITS AND PHYSICAL CONSTANTS. [CHAP. The second equation gives at once n = J. The third then gives m = - 1, and these values will be found to satisfy the first equation also. The velocity, then, varies directly as the square root of E, and inversely as the square root of D. 3. The frequency of vibration f for a musical string (that is, the number of vibrations per unit time) depends on its length I, its mass m, and the force with which it is stretched F. The dimensions of f are T" 1 . F MLT- 2 . Assume that f varies as l x m y ^ z . Then we have giving - 1 = - whence = VTT . 4. The angular acceleration of a uniform disc round its axis depends on the applied couple G, the mass of the disc M, and its radius R. Assume it to vary as G x M. y E, 2 . The dimensions of angular acceleration are T~' 2 . G ML 2 T- 2 . R I*. Hence we have m.] MECHANICAL UNITS. 37 giving - 2 = - 2x, x + y - 0, 2x + z - 0, whence x-1, ?/=], z-2. Hence the angular acceleration varies as 2 . JVL.LV In the following example the information obtained is less complete : 5. The range of a projectile on a horizontal plane through the point of projection depends on the initial velocity V, the intensity of gravity g, and the angle of elevation a. The dimensions of range are L. V T/T- 1 )) 5) 3) AJ - L 5) 5) 9 ?) * J * " ,, a, LT, and the dimensions of all powers of a are LT. Hence we can draw no inferences as to the manner in which a enters the expres- sion for the range. The dimensions of this expression will depend upon Y and g alone. Assume that the range varies as V m g n . Then T m-\-n HP m In . giving whence m-2, n - 1. "V 2 Hence the range varies as when a is given. J 38 CHAPTER IV, HYDROSTATICS. 34. THE following table of the relative density of water at various temperatures (under atmospheric pressure), the density at 4 C. being taken as unity, is from Rossetti's results deduced from all the best experiments (Ann. Ch. Phys. x. 461 ; xvii. 370, 1869) :- Temp. Cent. Relative Density. Temp. Cent. Relative Density. Temp. Cent. Relative Density. 999871 13 999430 35 99418 1 999923 14 999299 40 99235 2 999969 15 999160 45 99037 3 999991 16 999002 50 98820 4 1-000000 17 998841 55 98582 5 999990 18 998654 60 98338 6 999970 19 998460 65 98074 7 999933 20 998259 70 97794 8 999886 22 997826 75 97498 9 999824 24 997367 80 97194 10 999747 26 996866 85 96879 11 999655 28 996331 90 96556 12 999549 30 995765 100 95865 35. According to Kupffer's observations, as reduced by Professor W. H. Miller, the absolute density (in grammes per cubic centimetre) at 4 is not 1, but 1-000013. Multiplying the above numbers by this CHAP. IV.] HYDROSTATICS. 39 factor, we obtain the following table of absolute den- sities : Temp Density. Temp. Density. | Temp. Density. 6 999884 13 999443 35 99469 i 999941 14 999312 40 99236 2 999982 15 999173 45 99038 3 1-000004 16 999015 50 98821 4 1-000013 17 998854 55 98583 5 1-000003 18 998667 60 98339 6 999983 19 998473 65 98075 7 999946 20 998272 70 97795 8 999899 22 997839 75 97499 9 999837 24 997380 80 97195 10 999760 26 996879 85 96880 11 999668 28 996344 90 96557 12 999562 30 995778 100 95866 36. The volume, at temperature t\ of the water which occupies unit volume at 4, is approximately 1 + A(*-4) 2 -B(-4) 2 - 6 + C(*-4) 3 , where A = 8-38 x 10- 6 , B = 3-79 x 10- 7 , C = 2-24 x 10- 8 ; and the relative density at temperature t is given by the same formula with the signs of A, B, and C reversed. The rate of expansion at temperature t is In determining the signs of the terms with the frac- tional exponents 2*6 and 1*6, these exponents are to be regarded as odd. 37. The following Table of Densities has been compiled by collating the best authorities, but is only to be taken 40 UNITS AND PHYSICAL CONSTANTS. [CHAP. as giving rough approximations. Most of the densities vary between wide limits in different specimens : Solids. Aluminium, Antimony, 2-6 6-7 Carbon (diamond),., (graphite),.. ,, (gas carbon), ,, (wood charcoal), Phosphorus (ordi- nary), (red),... Sulphur (roll), Quartz (rock cry- stal), Sand (dry), Clay, . 3-5 2-3 1-9 1-6 1-83 2-2 2-0 2-65 1-42 1-9 2-1 3-0 8 to 2-8 5 to 2-7 to 3-5 2-4 Bismuth 9'8 Brass, 8-4 Copper, Gold, 8-9 19-3 Iron, 7'8 Lead, . . . 11 -8 Nickel, Platinum, 8-9 21-5 Silver, 10-5 Sodium, . . 98 Tin 7'3 Brick, Basalt Zinc 7'1 Cork, 24 Chalk 1 Oak, .. . 7 to 1 '0 Glass (crown), 2 ,, (flint), 3 Ebony, Ice. .'. l-ltol-2 918 Porcelain . . . 1 -026 8 1-5 73 Sulphuric Acid,... . Nitric Acid, Hydrochloric Acid, Milk, 1-85 1-56 1-27 1-03 1-29 1-27 3'5'JG Oil of Turpentine,.. ,, Linseed, ,, Mineral, 87 94 76 to -83 Liquids at C. Sea water, Alcohol, Chloroform, Ether, Bisulphide of Carbon,.. Glycerine, T27 Mercury, 13'5 ( J6 More exactly, the density of mercury at C., as com- pared with water at the temperature of maximum density, under atmospheric pressure, is 13*5956. 38. If a body weighs m grammes in vacuo and ra' grammes in water of density unity, the volume of the body is m - m' cubic centims. ; for the mass of the water displaced is m - m' grammes, and each gramme of this water occupies a cubic centimetre. TV.] HYDROSTATICS. 41 Examples. 1. A glass cylinder, I centims. long, weighs m grammes in vacuo and m' grammes in water of unit density. Find its radius. Solution. Its section is Trr 2 , and is also m ~ m ; hence L o m m! r 2 = Trl 2. Find the capacity at C. of a bulb which holds m grammes of mercury at that temperature. Solution. The specific gravity of mercury at being 13 '5 95 6 as compared with water at the temperature of maximum density, it follows that the mass of 1 cubic centirn. of mercury is 13-5956 x 1-000013 = 13-5958, say 13 '5 9 6. Hence the required capacity is cubic lo '0\y O centims. 3. Find the total pressure on a surface whose area is A square centims. when its centre of gravity is immersed to a depth of h centims. in water of unity density, atmos- pheric pressure being neglected. Ans. A.h grammes weight ; that is, gAk dynes. 4. If mercury of specific gravity 13-596 is substituted for water in the preceding question, find the pressure. Ans. 13-596 AJi grammes weight ; that is, 13*596 gAJi dynes. 5. If h be 76, and A be unity in example 4, the answer becomes 1033*3 grammes weight, or 1033 -3g dynes. For Paris, where g is 980-94, this is I'0136xl0 6 dynes. 42 UNITS AND PHYSICAL CONSTANTS. [CHAP. Harometric Pressure. 39. The C.G.S. unit of pressure intensity (that is, of pressure per unit area) is the pressure of a dyne per square centim. At the depth of h centiins. in a uniform liquid whose density is d [grammes per cubic centim.], the pressure due to the weight of the liquid is ghd dynes per square centim. The pressure-intensity due to the weight of a column of mercury at C., 76 centhns. high, is found by putting &=76, d= 13*596, and is 1033%. It is therefore different at different localities. At Paris, where g is 980-94, it is 1-0136 x 10 6 ; that is, rather more than a megadyne * per square centim. To exert a pressure of exactly one megadyne per square centim., the height of the column at Paris must be 74*98 centims. At Greenwich, where g is 981-17, the pressure due to 76 centims. of mercury at C. is 1*0138 x 10 6 ; and the height which would give a pressure of 10 6 is 74*964 centims., or 29-514 inches. Convenience of calculation would be promoted by adopting the pressure of a megadyne per square centim., or 10 6 C.G.S. units of pressure-intensity, as the standard atmosphere. The standard now commonly adopted (whether 76 centims. or 30 inches) denotes different pressures at different places, the pressure denoted by it being pro- portional to the value of g. We shall adopt the megadyne per square centim. as our standard atmosphere in the present work. *The prefix mega denotes multiplication by a million. A megadyne is a force of a million dynes. iv.] HYDROSTATICS. 4 a Examples. 1. What must be the height of a column of water of unit density to exert a pressure of a megadyne per square centim. at a place where g is 981 1 Ans. 1000Q00 - 1019-4 centims. This is 33-445 feet. i/o JL 2. What is the pressure due to an inch of mercury at C. at a place where g is 981 1 (An inch is 2*54 centims.) Ans. 981 x 2-54 x 13*596 = 33878 dynes per square centim. 3. What is the pressure due to a centim. of mercury at C. at the same locality ? Ans. 981 x 13-596 = 13338. 4. What is the pressure due to a kilometre of sea- water of density 1-027, g being 981 1 Ans. 981 x 10 5 x 1 -027 = 1'0075 x 10 s dynes per square centim., or 1-0075 x 10 2 megadynes per square centim. ; that is, about 100 atmospheres. 5. What is the pressure due to a mile of the same water 1 Ans. 1-6214 x 10 8 C.G.S. units, or 162-14 atmospheres [of a megadyne per square centim.]. Density of Air. 40. Regnault found that at Paris, under the pressure of a column of mercury at 0, of the height of 76 centims., the density of perfectly dry air was '0012932 gramme per cubic centim. The pressure corresponding to this height of the barometer at Paris is 1 -0136 x 10 6 dynes per square 44 UNITS AND PHYSICAL CONSTANTS. [CHAP- centini. Hence, by Boyle's law, we can compute the density of dry air at 0. at any given pressure. At a pressure of a megadyne (10 6 dynes) per square centini. the density will be . . " = -0012759. 1 "Olob The density of dry air at C. at any pressure p (dynes per square centim.) is PY. l-2759x!0- 9 .... (4) Example. Find the density of dry air at C., at Edinburgh, under the pressure of a column of mercury at C., of the height of 76 centims. Here we have p = 981 -54 x 76 x 13-596 = 1-0142 x 10 6 . Ans, Required density = 1-2940 x 10~ 3 = -0012940 gramme per cubic centim. 41. Absolute Densities of Gases, in grammes per cubic centim., at C., and a pressure of 10 6 dynes per square centim. Mass of a cubic Volume of a gramme centim. in grammes, in cubic centims. Air, dry, .................. -0012759 ......... 783"S Oxygen, .................... "0014107 ......... 70S"9 Nitrogen, ................... -0012393 ......... 806"9 Hydrogen, ................. "00008837 ........ 11316-0 Carbonic Acid, ............ "0019509 ......... 512-6 Oxide, .......... -0012179 ......... 821-1 Marsh Gas, ................ "0007173 ......... 1394-1 Chlorine, .................... "0030909 ......... 323'5 Protoxide of Nitrogen,.. "0019433 ......... 514-6 Binoxide ,, ... "0013254 ......... 754-5 Sulphurous Acid, ........ '0026990 ......... 370-5 Cyanogen, .................. "0022990 ......... 435-0 OlefiantGas, ............... -0012529 ......... 798-1 Ammonia,.. "0007594 1316 "8 iv.] HYDROSTATICS. 45 The numbers in the second column are the reciprocals of those in the first. The numbers in the first column are identical with the specific gravities referred to water as unity. Assuming that the densities of gases at given pressure and temperature are directly as their atomic weights, we have for any gas at zero pvp= 1-1316 xl0 10 m; v denoting its volume in cubic centims., m its mass in grammes, p its pressure in dynes per square centim., and /A its atomic weight referred to that of hydrogen as unity. Height of Homogeneous Atmosphere. 42. We have seen that the intensity of pressure at depth 7i, in a fluid of uniform density d, is ghd when the pressure at the upper surface of the fluid is zero. The atmosphere is not a fluid of uniform density ; but it is often convenient to have a name to denote a height H such that p = #HD, where p denotes the pressure and D the density of the air at a given point. It may be defined as the height of a column of uniform fluid having the same density as the air at the point, which would exert a pressure equal to that existing at the point. If the pressure be equal to that exerted by a column of mercury of density 13'596 and height h, we have p=ghx 13-596; .-. HD = A x 13-596, H = Ax18 ' 596 . If it were possible for the whole body of air above the point to be reduced by vertical compression to the density which the air has at the point, the height from the point 46 UNITS AND PHYSICAL CONSTANTS. [CHAP. up to the summit of this compressed atmosphere would be equal to H, subject to a small correction for the variation of gravity with height. H is called the height of the homogeneous atmosphere at the point considered. Pressure-height would be a better name. The general formula for it is H-; ... (5) and this formula will be applicable to any other gas as well as dry air, if we make D denote the density of the gas (in grammes per cubic centim.) at pressure p. If, instead of p being given directly in dynes per square centim., we have given the height h of a column of liquid of density d which would exert an equal pressure, the formula reduces to H = j (6) 43. The value of in formula (5) depends only on the nature of the gas and on the temperature ; hence, for a given gas at a given temperature, H varies inversely as g only. For dry air at zero we have, by formula (4), 7-8376 x 10 8 1 = - . 9 At Paris, where g is 9 80 '9 4, we find H = 7-990 xlO 5 . At Greenwich, where g is 981-17, H = 7-988 xlO 5 . iv.] HYDROSTATICS. 47 Examples. 1. Find the height of the homogeneous atmosphere at Paris for dry air at 10 C., and also at 100 C. Ans. For given density, p varies as 1 x -00366 t, t de- noting the temperature on the Centigrade scale. Hence we have, at 10 C., H - 1-0366 x 7-99 x 10 5 = 8-2825 x 10 5 ; and at 100 C., H = 1-366 x 7-99 x 10 5 - 1-0914 x 10 6 . 2. Find the height of the homogeneous atmosphere for hydrogen at 0, at a place where g is 981. Here we have Diminution of Density with increase, of Height in the Atmosphere. 44. Neglecting the variation of gravity with height, the variation of H as we ascend in the atmosphere would depend only on variation of temperature. In an atmos- phere of uniform temperature H will be the same at all heights. In such an atmosphere, an ascent of 1 centim. will involve a diminution of the pressure (and therefore of the density) by of itself, since the layer of air which has been traversed is -- of the whole mass of superincum- H bent air. The density therefore diminishes by the same fraction of itself for every centim. that we ascend; in other words, the density and pressure diminish in geo- metrical progression as the height increases in arithmetical progression. 48 UNITS AND PHYSICAL CONSTANTS. [CHAP. Denote height above a fixed level by x, and pressure by p. Then, in the notation of the differential calculus, we have $=-^, II p and if p lt p 2 are the pressures at the heights x lt x z , we deduce x 2 - a?! - H log.Pl = H x 2-3026 log TO &. . . (7) Pz Pz In the barometric determination of heights it is usual to compute H by assuming a temperature which is the arithmetical mean of the temperatures at the two heights. For the latitude of Greenwich formula (7) becomes a? 2 - a^ = (1 x -00366 t) 7'988 x 10 5 x 2*3026 log^l P 2 = (1 x -00366 1) 1,839,300 log^l, . . (8) P2 t denoting the mean temperature, and the logarithms being common logarithms. To find the height at which the density would be halved, variations of temperature being neglected, we must put 2 for O in these formulae. The required height will be H Pi log e 2, or, in the latitude of Greenwich, for temperature 0., will be 1-8393 x 10 6 x -30103 - 553700. The value of log e 2, or 2-3026 Iog 10 2, is 2-3026 x -30103- -69315. Hence for an atmosphere of any gas at uniform tempera- ture, the height at which the density would be halved is the height of the homogeneous atmosphere for that gas, multiplied by '69315. The gas is assumed to obey Boyle's law. HYDROSTATICS. 49 Examples. 1. Show that if the pressure of the gas at the lower station and the value of g be given, the height at which the density will be halved varies inversely as the density. 2. At what height, in an atmosphere of hydrogen at C., would the density be halved, g being 98 1 1 ? Ana. 7-9954 x 10 6 . 45. The phenomena of capillarity, soap-bubbles, etc., can be reduced to quantitative expression by assuming a tendency in the surface of every liquid to contract. The following table exhibits the intensity of this contractile force for various liquids at the temperature of 20 C. The contractile force diminishes as the temperature in- creases. Superficial tensions at 20 C., in dynes per linear centim., deduced from Quincke's results. Tension of Surface separating the Liquid from Density. Air. Water. Mercury. Water, 0-9982 81 418 Mercury, 13-5432 540 418 Bisulphide of Carbon, 1-2687 32-1 41-75 372-5 Chloroform, 1-4878 30-6 29-5 399 Alcohol, 7906 25-5 399 Olive Oil, 9136 36-9 20 : 56 335 Turpentine, 8867 29-7 11-55 250-5 Petroleum, 7977 31-7 27'8 284 Hydrochloric Acid, 1-1 70-1 377 Solution of Hyposul- ) phite of Soda, - - ) 1-1248 77-5 442-5 The values here given for water and mercury are only applicable when special precautions are taken to ensure 50 UNITS AND PHYSICAL CONSTANTS. [CHAP. cleanliness and purity. Without such precautions smaller values will be obtained. (Quincke in Wied. Ann., 1886, page 219.) The following values are from the observations of A. M. Worthington (Proc. Roy. Soc., June 16, 1881), at tempera- tures from 15 to 18 C., for surfaces exposed to air : Surface Tension. In gm. per cm. In dynes per cm. Water, '072 to '080 70'6 to 78'5 Alcohol, -02586 25'3 Turpentine, '02818 27 '6 Olive Oil, -03373 33'1 Chloroform, '03025 29 "6 46. Very elaborate measurements of the thicknesses of soap films have been made by Reinold and Riicker (Phil. Trans., 1881, p. 456 ; and 1883, p. 651). When so thin as to appear black, the thickness varied from 7 "2 to 14 -5 millionths of a millimetre, the mean being 11 '7. This is 1*17 x 10~ 6 centimetre. The following thicknesses were observed for the colours of the successive orders : Thickness. cm. FIRST ORDER Red, 2'84xlO- 6 SECOND ORDER Violet, 3'05 Blue, 3-53 Green, 4'09 Yellow, 4-54 Orange, 4'91 Red, 5'22 THIRD ORDER Purple, 5'59 Blue, 5'77 ,, 6-03 Green, 6'56 Thickness. cm. Yellow, 7.10xlO- 5 Red, 7-65 Bluish Red, 8'15 ,, FOURTH ORDER Green, 8'41 ,, 8 '93 , , Yellow-Green,.. 9 '64 ,, Red, 10-52 FIFTH ORDER Green, M19xlO- 4 1-188 , Red,, 1-260 1-335 iv.] HYDROSTATICS. 51 Thickness, cm. SIXTH ORDER Green, l'410x!0- 4 1-479 Red, 1-548 1-627 ! SEVENTH ORDER Green,.... 1 705 Thickness, cm. Green, 1787 xlO~ 4 Red, 1-869 1-936 EIGHTH ORDER Green, 2'004 ,, Red,.. , 2-115 , 4 6 A. Depression of the barometrical column due to capillarity, according to Pouillet : Internal Diameter of tube. 2 2-5 3 3-5 4 4-5 5 5-5 6 6-5 7 7'5 8 Internal Depression. Diameter of tube. mm. mm. 4-579 8-5 3-595 9 2-902 9-5 2-415 10 2-053 10-5 1-752 11 1-507 11-5 1-306 12 1-136 12-5 995 13 877 13-5 775 14 684 14-5 Internal Depression. Diameter of tube. mm. mm. 604 15 534 15-5 473 16 419 16-5 372 17 330 17-5 293 18 750 18-5 230 19 204 19-5 181 20 161 20-5 143 21 Depression, mm. 127 112 099 087 077 068 060 053 047 041 036 032 028 52 CHAPTER V. STRESS, STRAIN, AND RESILIENCE. 47. IN the nomenclature introduced by Rankine, and adopted by Thomson and Tait, any change in the shape or size of a body is called a strain, and an action of force tending to produce a strain is called a stress. We shall always suppose strains to be small ; that is, we shall sup- pose the ratio of the initial to the final length of every line in the strained body to be nearly a ratio of equality. 48. A strain changes every small spherical portion of che body into an ellipsoid ; and the strain is said to be homogeneous when equal spherical portions in all parts of the body are changed into equal ellipsoids with their corresponding axes equal and parallel. When the strain consists in change of volume, unaccompanied by change of shape, the ellipsoids are spheres. When strain is not homogeneous, but varies continu- ously from point to point, the strain at any point is defined by attending to the change which takes place in a very small sphere or cube having the point at its centre, so small that the strain throughout it may be regarded as homogeneous. In what follows we shall suppose strain to be homogeneous, unless the contrary is expressed. CHAP, v.] STRESS, STRAIN, AND RESILIENCE. 53 49. The axes of a strain are the three directions in the body, at right angles to each other, which coincide with the directions of the axes of the ellipsoids. Lines drawn in the body in these three directions will remain at right angles to each other when the body is restored to its unstrained condition. A cube with its edges parallel to the axes will be altered by the strain into a rectangular parallelepiped. Any other cube will be changed into an oblique parallele- piped. When the axes have the same directions in space after as before the strain, the strain is said to be unaccompanied by rotation. When such parallelism does not exist, the strain is accompanied by rotation, namely, by the rotation which is necessary for bringing the axes from their initial to their final position. The numbers which specify a strain are mere ratios, and are therefore independent of units. 50. When a body is under the action of forces which strain it, or tend to strain it; if we consider any plane section of the body, the portions of the body which it separates are pushing each other, pulling each other, or exerting some kind of force upon each other, across the section, and the mutual forces so exerted are equal and opposite. The specification of a stress must include a specification of these forces for all sections, and a body is said to be homogeneously stressed when these forces are the same in direction and intensity for all parallel sec- tions. We shall suppose stress to be homogeneous, in what follows, unless the contrary is expressed. 51. When the force-action across a section consists of 54 UNITS AND PHYSICAL CONSTANTS. [CHAP. a simple pull or push normal to the section, the direction of this simple pull or push (in other words, the normal to the section) is called an axis of the stress. A stress (like a strain) has always three axes, which are at right angles to one another. The mutual forces across a section not perpendicular to one of the three axes are in general partly normal and partly tangential one side of the sec- tion is tending to slide past the other. The force per unit area which acts across any section is called the intensity of the stress on this section, or simply the stress on this section. The dimensions of "force per unit area," or - are , which we shall therefore call area LT 2 the dimensions of stress. 52. The relation between the stress acting upon a body and the strain produced depends upon the resilience of the body, which requires in general 21 numbers for its complete specification. When the body has exactly the same properties in all directions, 2 numbers are sufiicient. These specifying numbers are usually called coefficients of elasticity, but the word elasticity is used in so many senses that we prefer to call them coefficients of resilience. A coefficient of resilience expresses the quotient of a stress (of a given kind) by the strain (of a given kind) which it produces. A highly resilient body is a body which has large coefficients of resilience. Steel is an example of a body with large, and cork of a body with small, coefficients of resilience. In all cases (for solid bodies) equal and opposite strains (supposed small) require for their production equal and opposite stresses. v.] STRESS, STRAIN, AND RESILIENCE. 55 53. The coefficients of resilience most frequently re- ferred to are the three following : (1) Resilience of volume, or resistence to hydrostatic compression. If V be the original and V - v the strained volume, is called the compression, and when the body is subjected to uniform normal pressure P per unit over its whole surface, the quotient of P by the compres- sion is the resilience of volume. This is the only kind of resilience possessed by liquids and gases. (2) Young's modulus, or the longitudinal resilience of a body which is perfectly free to expand or contract laterally. In general, longitudinal extension produces lateral contraction, and longitudinal compression produces lateral extension. Let the unstrained length be L and the strained length L I, then is taken as the measure JL of the longitudinal extension or compression. The stress on a cross section (that is, on a section to which the stress is normal) is called the longitudinal stress, and Young's modulus is the quotient of the longitudinal stress by the longitudinal extension or compression. If a wire of cross section A sq. cm. is stretched with a force of F dynes, and its length is thus altered from L to L + 1, the value Tjl T of Young's modulus for the wire is . . A. L (3) " Simple rigidity " or resistance to shearing. This requires a more detailed explanation. 54. A shear may be denned as a strain by which a sphere of radius unity is converted into an ellipsoid of semiaxes 1, 1 + e, 1 - e ; in other words, it consists of an UNITS AND PHYSICAL CONSTANTS. [CHAP. extension in one direction combined with an equal com- pression in a perpendicular direction. 55. A unit square (Fig. 1) whose diagonals coincide with these directions is altered by the strain into a rhombus whose diagonals are (1 + e) ^2 and (1 - e) ^/2, and whose area, being half the product of the diagonals, is 1 - to the first order of small quantities, is 1, the same as the area of the original square. The length of a side of the rhombus, being the square root of the Fig.1 sum of the squares of the semi-diagonals, is found to be /\/l + e 2 or 1 + Je 2 , and is therefore, to the first order of small quantities, equal to a side of the original square. 56. To find the magnitude of the small angle which a side of the rhombus makes with the corresponding side of the square, we may proceed as follows : Let acb (Fig. 2) be an enlarged representation of one of the small tri- angles in Fig. 1 . Then we have ab = J, cb = \e ^2 = ', A./ 2 angle cba = -. Hence the length of the perpendicular cd is cb sin-= -- -- =^; and since ad is ultimately 4 <s/2 \/2 2 equal to ab, we have, to the first order of small quan- tities, v.] STRESS, STRAIN, AND RESILIENCE. 57 angle cab= cd -^ e = e. ad \ The semi-angles of the rhombus are therefore e, and the angles of the rhombus are - '2e ; in other words, each angle of the square has been altered by the amount 2e. This quantity '2e is adopted as the measure of the shear. 57. To find the perpendicular distance between oppo- site sides of the rhombus, we have to multiply a side by the cosine of 20, which, to the first order of small quan- tities, is 1. Hence the perpendicular distance between opposite sides of the square is not altered by the shear, and the relative movement of these sides is represented Fiff 3 - Fig .4 by supposing one of them to remain fixed, while the other slides in the direction of its own length through a distance of 2e, as shown in Fig. 3 or Fig. 4. Fig. 3, in fact, represents a shear combined with right-handed rota- tion, and Fig. 4 a shear combined with left-handed rota- tion, as appears by comparing these figures with Fig. 1, which represents shear without rotation. 58. The square and rhombus in these three figures may be regarded as sections of a prism whose edges are per- pendicular to the plane of the paper, and figures 3 and 4 58 UNITS AND PHYSICAL CONSTANTS. [CHAP. show that (neglecting rotation) a shear consists in the relative sliding of parallel planes without change of dis- tance, the amount of this sliding being proportional to the distance, and being in fact equal to the product of the distance by the numerical measure of the shear. A good illustration of a shear is obtained by taking a book, and making its leaves slide one upon another. It may be well to remark, by way of caution, that the selection of the planes is not arbitrary as far as direction is concerned. The only planes which are affected in the manner here described are the two sets of planes which make angles of 45 with the axes of the shear (these axes being identical with the diagonals in Fig. 1). 59. Having thus defined and explained the term " shear," which it will be observed denotes a particular species of strain, we now proceed to define a shearing stress. A shearing stress may be defined as the combination of two longitudinal stresses at right angles to each other, these stresses being opposite in sign and equal in magni- tude ; in other words, it consists of a pull in one direction combined with an equal thrust in a D c perpendicular direction. 60. Let P denote the intensity of each of these longitudinal stresses; we shall proceed to cal- culate the stress upon a plane in- -3 clined at 45 to the planes of these stresses. Consider a unit cube so taken that the pull is perpendicular to two of its faces, AB and DC (Fig. 5), and the thrust v.] STRESS, STRAIN, AND RESILIENCE. 5i> is perpendicular to two other faces, AD, BC. The forces which hold the half-cube ABC in equilibrium are (1) An outward force P, uniformly distributed over the face AB, and having for its resultant a single force P acting outward applied at the middle point of AB. (2) An inward force P, having for its resultant a single force P acting inwards at the middle point of BC. (3) A force applied to the face AC. To determine this third force, observe that the other two forces meet in a point, namely, the middle point of AC, that their components perpendicular to AC destroy one another, and that their components along AC, or p rather along CA, have each the magnitude - - . ; hence v^ their resultant is a force P ^2, tending from towards A. The force (3) must be equal and opposite to this. Hence each of the two half-cubes ABC, ADC exerts upon the other a force P ^2, which is tangential to their plane of separation. The stress upon the diagonal plane AC is therefore a purely tangential stress. To compute its intensity we must divide its amount P ^2 by the area of the plane, which is ^/2, and we obtain the quotient P. Similar reasoning applies to the other diagonal plane BD. P is taken as the measure of the shearing stress. The above discussion shows that it may be defined as the intensity of the stress either on the planes of purely normal stress, or on the planes of purely tangential stress. 61. A shearing stress, if applied to a body which has the same properties in all directions (an isotropic body), produces a simple shear with the same axes as the stress ; for the extension in the direction of the pull will be equal to the compression in the direction of the thrust ; and in 60 UNITS AND PHYSICAL CONSTANTS. [CHAP. the third direction, which is perpendicular to both of these, there is neither extension nor contraction, since the transverse contraction due to the pull is equal to the transverse extension due to the thrust. A shearing stress applied to a body which has not the same properties in all directions produces in general a shear with the same axes as the stress, combined with some other distortion. In both cases, the quotient of the shearing stress by the shear produced is called the resistance to shearing. In the case of an isotropic body, it is also called the simple rigidity. 62. The following values of the resilience of liquids under compression are reduced from those given in Jamin, "Cours de Physique," 2nd edition, torn. i. pp. 168 and 169: Temp Cent. Coefficient of Resilience. Compression for one Atmosphere (megadyne per square centim.) Mercury, - o-o 3'436xlO n 2-91 x 10- 6 Water, o-o 2-02 xlO 10 4-96 xlO-" 5 . 1-5 1-97 5-08 , . 4-1 2-03 4-92 . 10-8 2-11 4-73 13-4 2-13 4-70 , . 13-0 2-20 4-55 , . 25-0 222 4-50 34-5 224 4-47 . 43-0 2-29 4-36 , . 53-0 2-30 4-35 , Ether, - ( ') o-o \ U4-OJ 9'2 x 10 9 7-8 7'2 l-09x!0- 4 1-29 , 1-38 Alcohol, \ 7-3 | j 13-1 i 1-22 xlO 10 1-12 8-17 S-91 x 10-- r > Sea Water, - 17'5 2-33 4-30 ,, STRESS, STRAIN, AND RESILIENCE. 61 63. The following are reduced from the results ob- tained by Amaury and Descamps, " Comptes Rendus," torn. Ixviii. p. 1564 (1869), and are probably more accurate than the foregoing, especially in the case of mercury : Coefficient of Resilience. Compression for one megadyne per square centim. Distilled Water, 15 2'22 x 10 10 4-51 x 10- s Alcohol, 1-21 8-24 ,, . 15 I'll ,, 8-99 ,, Ether, 9'30xl0 9 1-OSxlO- 1 14 7-92 1-26 Bisulphide of Carbon, Mercury, - 14 15 l-60xl0 10 5'42xlO n 6-26xlO- 5 l-84x!0- 6 64. The following values of the coefficients of resilience for solids are reduced from those given in my own papers to the Royal Society (see " Phil. Trans.," Dec. 5th, 1867, p. 369), by employing the value of g at the place of ob- servation, namely, 981 '4. Young's Modulus, Simple Rigidity. Resilience of Volume, Density Glass, flint, 6-03 xlO 11 2-40 x 10 11 4-15 xlO 11 2-942 Another specimen 5-74 2-35 3-47 2-935 Brass, drawn, - 1-075X10 1 - 3-66 8-471 Steel, - 2-139 ,, 8-19 1 -841x1 1 - 7'849 Iron, wrought, - 1-963 ,, 7-69 1-456 7-677 cast,- 1-349 5-32 9-64 xlO 11 7-235 Copper, - 1-234 4-47 l-684x!0 12 8-843 65. The resilience of volume was not directly observed, but was calculated from the values of " Young's modulus " and " simple rigidity," by a formula which is strictly true 62 UNITS AND PHYSICAL CONSTANTS. [CHAP. for bodies which have the same properties in all direc- tions. The contraction of diameter in lateral directions for a body which is stretched by purely longitudinal stress was also calculated by a formula to which the same remark applies. The ratio of this lateral contraction to the longitudinal extension is called " Poisson's ratio," and the following were its values as thus calculated for the six bodies experimented on : Glass, flint, '258 Iron, wrought, '275 Another specimen, "229 ,, cast, '267 Brass, drawn, '469 (?) Copper, '378 Steel, '310 Kirchhoff has found for steel the value '294, and Clerk Maxwell has found for iron -267. Cornu ("Cornptes Rendus," August 2, 1869) has found for different speci- mens of glass the values -225, -226, '224, -257, '236, '243, 250, giving a mean of '237, and maintains (with many other continental savants) that for all iso tropic solids (that is, solids having the same properties in all direc- tions) the true value is J. 66. The following values of Poisson's ratio have been found by Mr. A. Mallock (" Proc. Roy. Soc.," June 19, 1879) : Steel .... '253 Ivory, .. -50 .. '50 50 Brass 325 India Rubber, Paraffin, Copper, 348 Lead, . . ... .... '375 Plaster of Paris,. . Cardboard, Cork, ... .. -181 .. -2 00 Zinc (rolled),.... (CcLSt) .... '180 230 Ebonite, . '389 Lial Longitudinal s to due to ndinal. Radial. 2 -406 3 '408 86 '372 In Cross Section, 227 Boxwood. Ra< due Longit Beechwood, White Pine, 5 -4 v.] STRESS, STRAIN, AND RESILIENCE. 63 The heading "Radial due to Longitudinal" means that the applied force is longitudinal (that is, parallel to the length of the tree) and that the contraction along a radius of the tree is compared with the longitudinal extension. 67. The following are reduced from Sir W. Thomson's results ("Proc. Roy. Soc.," May, 1865), the value of g being 981 '4: Simple Rigidity. Brass, three specimens, 4'03 3*48 3'44 ) 1A11 r* - J_ ' . A.Af\ A.4f\ f X 1U Copper, two specimens, 4*40 4*40 Other specimens of copper in abnormal states gave results ranging from 3'86 x 10 11 to 4'64 x 10 11 . The following are reduced from Wertheirn's results ("Ann. de Chim.," ser. 3, torn, xxiii.), g being taken as 981: Different Specimens of Glass (Crystal}. Young's Modulus, 3 '41 to 4 '34, mean 3 '96 ^ Simple Rigidity, 1 -26 to 1 '66 1'48 [ x 10 11 Volume Resilience 3'50 to 4*39 ,, 3'89j Different Specimens of Brass. Young's Modulus,.... 9*48 to 10 '44, mean 9 '86 \ Simple Rigidity, 3 '53 to 3 '90 ,, 3 '67 [ x 10 11 Volume Resilience,.. 10'02tol0'85 ,, 10'43 J 68. Savart's experiments on the torsion of brass wire (" Ann. de Chim.," 1829) lead to the value 3*61 x 10 11 for simple rigidity. Kupffer's values of Young's modulus for nine different specimens of brass range from 7 '9 6 x 10 11 to 11 '4 x 10 11 , the value generally increasing with the density. For a specimen, of density 8*4465, the value was 10-58 x 10 11 . 64 UNITS AND PHYSICAL CONSTANTS. [CHAP. v. For a specimen, of density 8 '4930, the value was 11-2 x 10 11 . The values of Young's modulus found by the same experi- menter for steel, range from 20 -2 x 10 11 to 21 -4 x 10 11 . 69. The following are reduced from Eankine's " Kules and Tables," pp. 195 and 196, the mean value being adopted where different values are given : Steel Bars, Tenacity. 7'93x 10 9 Y< >ung's Modulus. 2-45 x 10 13 Iron Wire, 5-86 ,, 1 '745 . Copper Wire, Brass Wire, Lead, Sheet, Tin, Cast, 4-14 3-38 2-28 x 10 8 3-17 1-172 9-81 x!0 1] 5-0 xlO 10 Zinc, 5-17 ,, Ash,. ri72xlO !) 1-10 x 10 11 Spruce, Oak, Glass, Brick and Cement.. . 8-55 xlO 8 1 -026x10'' 6-48 x 10 s 2-0 xl()< 1-10 ,, 1-02 5-52 xlO 11 The tenacity of a substance may be defined as the greatest longitudinal stress that it can bear without tear- ing asunder. The quotient of the tenacity by Young's modulus will therefore be the greatest longitudinal exten- sion that the substance can bear. 65 CHAPTER VI. ASTRONOMY. Size and Figure of the Earth. 70. ACCORDING to the latest determination, as pub- lished by Capt. Clarke in the " Philosophical Magazine " for August, 1878, the semiaxes of the ellipsoid which most nearly agrees with the actual earth are, in feet, a = 20926629, b = 20925105, c = 20854477, which, reduced to centimetres, are a = 6-37839 x 10 8 , b = 6-37792 x 10 s , c = 6-35639 x 10 8 , giving a mean radius of 6 '3709 x 10 8 , and a volume of 1-0832 x 10 27 cubic centims. The ellipticities of the two principal meridians are 289-5 and 291T8' The longitude of the greatest axis is 8 15' W The mean length of a quadrant of the meridian is 1*00074 x 10 9 . The length of a minute of latitude is approximately 185200 - 940 cos. 2 lat. of middle of arc. The mass of the earth, assuming Baily's value 5*67 for the mean density, is 6'14 x 10 27 grammes. E 66 UNITS AND PHYSICAL CONSTANTS. [CHAP. Day and Year. Sidereal day, ........................... 86164 mean solar seconds. Sidereal year, .......................... 31,558,150 Tropical year, ......................... 31,556,929 Angular velocity of earth's rotation, Velocity of earth in orbit, about 2960600 ,, Centrifugal force at equator due) O.OOAQ j to earth's rotation,.. ............. ) 3 3908 d y nes P er i ramme - Attraction, in Astronomy. 71. The mass of the moon is the product of the earth's mass by -011364, and is therefore to be taken as 6 '9 8 x 10 25 grammes, the doubtful element being the earth's mean density, which we take as 5 '67. The mean distance of the centres of gravity of the earth and moon is 60-2734 equatorial radii of the earth that is, 3-8439 x 10 10 centims. The mean distance of the sun from the earth is about 1-487 x 10 13 centims., or 92*39 million miles, correspond- ing to a parallax of 8"-848.* The intensity of centrifugal force due to the earth's motion in its orbit (regarded as circular) is I ") r, r de- noting the mean distance, and T the length of the sidereal year, expressed in seconds. This is equal to the accelera- tion due to the sun's attraction at this distance. Putting for r and T their values, 1-487 x 10 13 and 3-1558 x 10 7 , we have / \ r= -5894. * This value of the mean solar parallax was determined by Pro- fessor Newcomb, and was adopted in the " Nautical Almanac " for 1882. (See Art. 86 for a later determination.) vi.] ASTRONOMY. 67 This is about - - of the value of g at the earth's 1660 surface. The intensity of the earth's attraction at the mean dis- tance of the moon is about 981 or -2701. (60-27) 2 This is less than the intensity of the sun's attraction upon the earth and moon, which is "5894 as just found. Hence the moon's path is always concave towards the sun. 72. The mutual attractive force F between two masses m and m', at distance I, is T? _ fl mm ' where C is a constant. To determine its value, consider the case of a gramme at the earth's surface, attracted by the earth. Then we have whence we find Cr= 6j48 = 1 10 8 l-543x!0 7 ' To find the mass m which, at the distance of 1 centim. from an equal mass, would attract it with a force of 1 dyne, we have 1 = C/n 2 ; whence m = I - = 3928 grammes. 73. To find the acceleration a produced at the distance of I centims. by the attraction of a mass of m grammes, we have a = = C-, m' P where C has the value 6*48 x 10~ 8 as above. 68 UNITS AND PHYSICAL CONSTANTS. [CHAP. To find the dimensions of C we have C = , where the m dimensions of a are LT~ 2 . The dimensions of C are therefore L 2 M- 1 LT~ 2 ; that is, L 3 M' 1 T~ s . 74. The equation = C^ shows that when a = l and 1=1, m must equal ; that is to say, the mass which O produces unit acceleration at the distance of 1 centimetre is 1*543 x 10 7 grammes. If this were taken as the unit of mass, the centimetre and second being retained as the units of length and time, the acceleration produced by the attraction of any mass at any distance would be simply the quotient of the mass by the square of the distance. It is thus theoretically possible to base a general system of units upon two fundamental units alone ; one of the three fundamental units which we have hitherto employed being eliminated by means of the equation mass = acceleration x (distance) 2 , which gives for the dimensions of M the expression L 3 T- 2 . Such a system would be eminently convenient in astro- nomy, but could not be applied with accuracy to ordinary terrestrial purposes, because we can only roughly compare the earth's mass with the masses which we weigh in our balances. 75. The mass of the earth on this system is the product of the acceleration due to gravity at the earth's surface, and the square of the earth's radius. This product is 981 x (6-37 x!0 8 ) 2 = 3-98 x 10 20 , vi.] ASTRONOMY. 69 and is independent of determinations of the earth's density. The new unit of force will be the force which, acting upon the new unit of mass, produces unit acceleration. It will therefore be equal to 1'543 x 10 7 dynes; and its dimensions will be mass x acceleration = (acceleration) 2 x (distance) 2 = L 4 T -4 76. If we adopt a new unit of length equal to I centims., and a new unit of time equal to t seconds, while we define the unit mass as that which produces unit acceleration at unit distance, the unit mass will be Pt~- x 1-543 x 10 7 grammes. If we make I the wave-length of the line F in vacuo, say, 4-86 x lO" 5 , and t the period of vibration of the same ray, so that is the velocity of light in vacuo, say, 3 x 10 10 , the value of I s r' 2 or ill V fe 4-374 x 10 16 , and the unit mass will be the product of this quantity into 1-543 x 10 7 grammes. This product is 6-75 x 10 23 grammes. The mass of the earth in terms of this unit is 3-98 x 10 20 4- (4-374 x 10 16 ) = 9100, and is independent of determinations of the earth's density. 70 CHAPTER VII. VELOCITY OF SOUND. 77. THE propagation of sound through any medium is due to the elasticity or resilience of the medium ; and the general formula for the velocity of propagation s is E where D denotes the density of the medium, and E the coefficient of resilience. 78. For air, or any gas, we are to understand by E the quotient increment of pressure corresponding compression ' that is to say, if P, P + p be the initial and final pres- sures, and V, V - v the initial and final volumes, p and v being small in comparison with P and V, we have V If the compression took place at constant temperature, we should have But in the propagation of sound, the compression is effected so rapidly that there is not time for any sensible part of the heat of compression to escape, and we have CHAP. VIL] VELOCITY OF SOUND. 71 where y= 1-41 for dry air, oxygen, nitrogen, or hydrogen. p The value of =- for dry air at t Cent, (see p. 46) is (1 + -00366*) x 7-838 + 10 8 . Hence the velocity of sound through dry air is *= 10 4 ^1-41 x (1 + -00366*) x 7-838 = 33240^1 + -00366*; or approximately, for atmospheric temperatures, = 33240 + 60*. 79. In the case of any liquid, E denotes the resilience of volume.* For water at 8-l C. (the temperature of the Lake of Geneva in Colladon's experiment) we have E = 2-08 x 10 10 , D = 1 sensibly ; -= > /E = 144000, D the velocity as determined by Colladon was 143500. 80. For the propagation of sound along a solid, in the form of a thin rod, wire, or pipe, which is free to expand or contract laterally, E must be taken as denoting Young's modulus of elasticity.* The values of E and D will be different for different specimens of the same material. Employing the values given in the Table ( 64), we have * Strictly speaking, E should be taken as denoting the resili- ence for sudden applications of stress so sudden that there is not time for changes of temperature produced by the stress to be sensibly diminished by conduction. This remark applies to both 79 and 80. For the amount of these changes of temperature, see a later section under Heat. 72 UNITS AND PHYSICAL CONSTANTS. [CHAP. Values of E. Values of Values of / , D. or velocity. Glass, first specimen, 6-03 xlO 11 2-942 4-53 x 10 5 ,, second specimen, 5-74 2-935 4-42 Brass, 1-075x1 1 12 8-471 3-56 Steel, 2-139 7-849 5-22 Iron, wrought, - 1-963 7-677 5-06 ,, cast, - 1-349 7-235 4-32 Copper, 1-234 ,, 8-843 3-74 81. If the density of a specimen of red pine be -5, and its modulus of longitudinal elasticity be T6 x 10 6 pounds per square inch at a place where g is 981, compute the velocity of sound in the longitudinal direction. By the table of stress, page 4, a pound per square inch (g being 981) is 6*9 x 10 4 dynes per square centim. Hence we have for the required velocity centims. per second. 82. The following numbers, multiplied by 10 5 , are the velocities of sound through the principal metals, as determined by Wertheim : At 20 C. At 100 C. At 200 C. Lead, 1-23 1-20 Gold, 1-74 1-72 1-73 Silver, 2-61 2-64 2-48 Copper, - Platinum, 3-56 2-69 3-29 2-57 2-95 2-46 Iron, 5-13 5-30 4-72 Iron Wire (ordinary), Cast Steel, 4-92 4-99 5-10 4-92 4-79 Steel Wire (English), 4-71 5-24 5-00 4-88 5-01 VII.] VELOCITY OF SOUND. 73 The following velocities in wood are from the observa- tions of Wertheim and Chevandier, " Comptes Rendus," 1846, pp. 667 and 668 : ., ., Radial Tangential Along Mbres. Direction . Direction. Pine, 3-32 xO 5 ! 2-83 xlO 5 1 I'59xl0 5 Beech, 3-34 3-67 i 2-83 Witch-E m, - 3-92 3-41 ,, 2-39 Birch, 4-42 2-14 ,, 3-03 ,, Fir, 4-64 2-67 1-57 Acacia, 4-71 Aspen, - 5-08 Musical Strings. 83. Let M denote the mass of a string per unit length, F stretching force, L ,, length of the vibrating portion ; then the velocity with which pulses travel along the string is F and the number of vibrations made per second is Example. For the four strings of a violin the values of M in grammes per centimetre of length are 00416, -00669, -0106, -0266. The values of n are 660, 440, 293J, 195$; 74 UNITS AND PHYSICAL CONSTANTS. [CHAP. VIK and the common value of L is 33 centims. Hence the values of v or 2Ln are 43560, 29040, 19360, 12910 centims. per second ; and the values of F or Mv 2 , in dynes, are 7-89 x 10 6 , 5-64 x 10 8 , 3-97 x 10 6 , 4-43 x 10 6 . Faintest Audible Sound. 84. Lord Rayleigh ("Proc. K. S.," 1877, vol. xxvi. p. 248), from observing the greatest distance at which a, whistle giving about 2730 vibrations per second, and blown by water-power, was audible without effort in the middle of a fine still winter's day, calculates that the maximum velocity of the vibrating particles of air at this distance from the source was '0014 centims. per second, and that the amplitude was 8'1 x 10~ 8 centims., the calculation being made on the supposition that the sound spreads uniformly in hemispherical waves, and no deduction being made for dissipation, nor for waste energy in blowing. 75 CHAPTER VIII. LIGHT. 85. ALL kinds of light are believed to have the same velocity in vacuo. The velocity of light of given re- frangibility in any medium is - of its velocity in vacuo, fji denoting the absolute index of refraction of that medium for light of the given refrangibility. Light of given refrangibility is light of given wave- frequency. Its wave-length in any medium is the quotient of its velocity in that medium by its wave- frequency. If n denote the wave-frequency (that is to say, the number of waves which traverse a given point in a second), the wave-length in any medium will be np of the velocity in vacuo. The absolute index of refraction for ordinary air is about 1*00029. More accurate statements of its value will be found in Arts. 94-96. 86. The best determination of the velocity of light is that made by Professor Newcomb at Washington in 1882 (" Astron. Papers of Amer. Ephem.," vol. ii. parts iii. and iv. 1885). The method employed was that of the revolving mirror, the distance between the revolving and 76 UNITS AND PHYSICAL CONSTANTS. [CHAP. the fixed mirror being in one portion of the observations 2550 metres, and in the remaining portion 3720 metres. The resulting velocity in vacuo is 2-99860 x 10 10 centims. per sec. The following summary of results is from Professor Newcomb's paper, page 202 : km. per. sec. Michelson, at Naval Academy, in 1879, 299910 Michelson, at Cleveland, 1882, 299853 Newcomb, at Washington, 1882, using only'j results supposed to be nearly free from 1- 299860 constant errors, J Newcomb, including all determinations, 299810 Foucault, at Paris, in 1862, 298000 Cornu, at Paris, in 1874, 298500 Cornu, at Paris, in 1878, 300400 This last result as discussed by Listing, 299990 Young and Forbes, 1880-81, 301382 Professor Newcomb remarks (page 203) that the value 299860 km. per sec. for the velocity of light, combined with Clark's value 6378-2 km. for the earth's equatorial radius, and Nyren's value 20"*492 for the constant of aberration, gives for the solar parallax the value 8" '7 94. 87. The following are the wave-lengths adopted by Angstrom for the principal Fraunhofer lines in air at 760 millims. pressure (at Upsal) and 16 C. : Centims. A 7-604 xlO~ 5 B 6-867 C 6-56201 ,, Mean of lines D 5-89212 ,, E 5-26913 F 4-86072 G 4-30725 H 5 3-96801 H 2 3-93300 V viii.] LIGHT. 77 These numbers will be approximately converted into the corresponding wave-lengths in vacuo by multiplying them by 1-00029. 88. Assuming 3 x 10 10 to be the velocity of light in air, and neglecting the difference of velocity between the more and less refrangible rays, we obtain the follow- ing frequencies by dividing the common velocity by Angstrom's values of the wave-lengths : Vibrations per Second. A 3-945 xlO 14 B 4-369 C 4-572 D 5-092 E 5-693 ,, F 6-172 G 6-965 H x 7-560 ,, H 2 7-628 According to Langley ("Com. Ren.," Jan., 1886), the solar spectrum extends beyond the red as far as wave- length 27 x 10~ 5 , and the radiation from terrestrial bodies at temperatures below 100 extends as far as wave-length 150 x 10~ 5 . The frequencies corresponding to these two wave-lengths are 1*1 x 10 14 and 2 x 10 13 . INDICES OF REFRACTION OF SOLIDS. 89. Dr. Hopkinson (" Proc. R. S.," June 14, 1877) has determined the indices of refraction of the principal varieties of optical glass made by Messrs. Chance, for the fixed lines A, B, C, D, E, 6, F, (G), G, h, H r By D is to be understood the more refrangible of the pair of sodium lines ; by b the most refrangible of the group of magnesium lines ; by (G) the hydrogen line near G. 78 UNITS AND PHYSICAL CONSTANTS. [CHAP. In connection with the results of observation, he employs the empirical formula /u,- 1 =a{l +bx(l + cx)}, where # is a numerical name for the definite ray of which ju is the refractive index. In assigning the value of x, four glasses hard crown, soft crown, light flint, and dense flint were selected on account of the good accord of their results ; and the mean of their indices for any given ray being denoted by /I, the value assigned to x for this ray is /i - /x p where /X F denotes the value of ji for the line F. The value of /x as a function of A., the wave-length in 10~ 4 centimetres, was found to be approximately ji. = 1-538414 + 0-00676601 - 0-00017341 + 0-0(X)023l. The following were the results obtained for the different specimens of glass examined : Hard Crown, 1st specimen, density 2 '48575. = 0'523145, = 1-3077, c= -2'33. Means of observed values of /u. A 1-511755; B 1-513624; C 1 '514571 ; D 1-517116; E 1 -520324 ; b 1 '520962 ; F 1 '523145 ; (G) 1 '527996 ; O 1-528348; h 1-530904; H x 1-532789. Soft Crown, density 2' 55035. a=0 -5209904, 6 = 1-4034, c=-l'58. Means of observed values of /A. A 1-508956; B 1-510918; C 1-511910; D 1-514580; E 1-518017; 61'518678; F 1-520994; (G) 1 '526208; G 1-526592; h 1 '529360; Hj 1 '531415. vm.] LIGHT. 79 Extra Light Flint Glass, density 2 '86636. a = 0'549123, 6 = 17064, c=-0'198. Means of observed values of /*. A 1-534067; B 1*536450; Cl '537682; D 1-541022; E 1 '545295; b 1 '546169; F 1-549125; (G) 1 '555870; G 1 '556375; 7i 1-559992; Hj 1 '5627 60. Light Flint Glass, density 3 '20609. a= 0-583887, 6=1-9605, c=0'53. Means of observed values of u. B 1-568558; E 1-579227; <G) 1-592184; H, 1-600717. C 1-570007; b 1-580273; G 1-592825; D 1-574013 F 1-583881 h 1-597332 Dense Flint, density 3 '65865. a = 0-634744, b = 2-2694, c = 1'48. Means of observed values of fi. B 1-615704; C 1-617477; D 1-622411 E 1-628882; b 1-630208; F 1-634748 <G) 1-645268; G 1 '64607 1 ; h 1 '651830 Extra Dense Flint, density 3 '88947. a=0'664226, 6=2-4446, c=l87. Means of observed values of /x. A 1-639143; B 1*642894; C 1-644871 D 1-650374; E 1-657631 ; b 1 '659 108 F 1-664246; (G) 1-676090; G 1 '677020 h 1-683575; H x 1-688590. 80 UNITS AND PHYSICAL CONSTANTS. [CHAP, Double Extra Dense Flint, density 4 '42162. a=0'727237, fc=27690, c=2'70. Means of observed values of p. A 1-696531; B 1-701080; Cl "703485; D 1-710224; E 1 "719081 ; b 1 '720908; F 1-727257; (G) 1742058; G 1-743210; h 1-751485. 90. The following indices of rock salt, sylvin, and alum for the chief Fraunhofer lines are from Stefan's observa tions : Rock Salt Sylvin Alum at 17 C. at 20 C. at 21 C. A 1-53663 1-48377 1-45057 B -53918 -48597 '45262 C -54050 -48713 -45359 D -54418 -49031 -45601 E -54901 -49455 -45892 F -55324 -49830 -46140 G -56129 -50542 -46563 H -56823 -51061 -46907 91. Indices of other singly refracting solids E Diamond, Index of .efraction. 2-470 1-4339 1-532 I 545 1-528 1-480 1-593 1-528 Kind of Light. D D D Red Red Red D Red Observer. Schrauf. Stefan. Kohlrausch. Jamin. ii > j Baden Powell. Wollaston. Fluor-spar, Amber, Rosin Copal, . Gum Arabic, Peru Balsam, Canada Balsam,. Effect of Temperature. According to Stefan, the index of refraction of glass increases by about -000002 for each degree Cent, of VIII.] LIGHT. 81 increase of temperature, and the index of rock salt diminishes by about '000 037 for each degree of increase of temperature. 92. Doubly refracting crystals : Uniaxal Crystals. Ice, Ordinary Index. 1 -3060 Extraordi- nary Index. 1-3073 Kind of Light. lied Tern] >. Observer. Reusch. Iceland-spar, Nitrate of Soda, Quartz, . 1-65844 1-5854 1-54419 1-48639 1 -3369 1-55329 D D D 24 23 24 v. d. Willigen. F. Kohlrausch. v. d. Willigen. Tourmaline,... . Zircon, .. 1-6479 1-92 1-6262 1-97 Green Red 22 Heusser. de Senarmont. Arragonite, Borax, Mica, Nitre, Selenite, Sulphur Biaxal Crystals. CIPAL INI Least. 1-53013 1-4463 T5609 1 -3346 1-52082 1-9505 (prismatic)/ Topaz, 1-61161 CES OF REFRACTION FOR SODIUM LIGHT. itermediate 1-68157 1-4682 i. Greatest. 1-68589 1-4712 Temp. 23 Observer. Rudberg. Kohlrausch. 1-5941 1-5997 23 ,, 1-5056 1 -5064 16 Schrauf. 1-52287 1-53048 17 v. Lang. 2-0383 2-2405 16 Schrauf. 1-61375 1-62109 Rudberg. INDICES OF REFRACTION FOR LIQUIDS. 93. The following values of indices of refraction for liquids are condensed from Fraunhofer's determinations, as given by Sir John Herschel (" Enc. Met. Art," Light, p. 415):- Water, density 1 '000. B 1-3309; C T3317 ; D P3336; E 1-3358; F1'3378; G1'3413; H 1-3442. 82 UNITS AND PHYSICAL CONSTANTS. [CHAP. Oil of Turpentine, density 0'885. B 1 -4705 ; C 1 -4715 ; D 1 "4744 ; E 1 '4784 ; F 1-4817; G 1-4882; H 1-4939. The following determinations of the refractive indices of liquids are from Gladstone and Dale's results, as given in Watt's "Dictionary of Chemistry," iii. pp. 629-631 : Sidphide of Carbon, at temperature 11. A 1-6142; B 1-6207; C 1'6240; D 1-6333; E 1-6465; Fl'6584; G 1-6836; H 17090. Benzene, at temperature 10 "5. A 1-4879; B 1-4913; C 1-4931 ; D 1-4975; E 1-5036; F 1-5089; G 1-5202; H 1-5305. Chloroform, at temperature 10. A 1-4438; B 1'4457; C 1-4466; D 1-4490; E 1-4526; F 1-4555; G 1-4614; H 1-4661. Alcohol, at temperature 15. A 1-3600; B 1-3612; C 1'3621 ; D 1*3638; E 1-3661; F 1-3683; Gl'3720; H 1-3751. Ether, at temperature 15. A 1-3529; B 1'3545; C 1-3554; D 1-3566; E 1-3590; F 1-3606; G 1-3646; H 1-3683. Water, at temperature 15. A 1 -3284 ; B 1 -3300 ; C 1 '3307 ; D 1 -3324 ; E 1-3347; F 1-3366; G 1*3402; H 1-3431. INDICES FOR GASES. 94. Indices of refraction of air at C. and 760 mm. for the principal fraunhofer lines. viii.] LIGHT. 83 According to Kettler. According to Lorenz. A 1-00029286 1-00028935 B 29350 28993 C 29383 29024 D 29470 29108 K 29584 29217 F 29685 29312 G 29873 29486 H 30026 29631 95. The formula established by the experiments of Biot and Arago for the index of refraction of air at various pressures and temperatures was _ -0002943 h l+at "760' a denoting the coefficient of expansion '00366, and h the pressure in millims. of mercury at zero. As the pressure of 760 millims. of such mercury at Paris is 1-0136 x 10 dynes per sq. cm., the general formula applicable to all localities alike will be _ l _ -0002943 P 1 + -00366* ' l-0136x!0 6 ' where P denotes the pressure in dynes per sq. cm. This can be reduced to the form jt _ 1 _ -0002903 _P 1 + -00366* ' 10* 96. According to Mascart, /JL - 1 for any gas is pro- portional not to - - but to h + /3k' 2 T+vT where /? and a are coefficients which vary from one gas to another. In the following table, the column headed /* 84 UNITS AND PHYSICAL CONSTANTS. [CHAP. contains the indices for and 760 mm. at Paris. The next column contains the value of /? multiplied by 10 7 (it being understood that h is expressed in millimetres), and the next column the value of a. All these data are for the light of a sodium flame : H> x 10 7 a' Air, 1-0002927 7 '2 '00382 Nitrogen, 2977 8'5 382 Oxygen, 2706 11-1 Hydrogen, 1387 -8'6 378 Nitrous Oxide, 5159 88 388 NitrousGas, 2975 7 367 Carbonic Oxide, 3350 8 '9 367 Carbonic Acid, 4544 72 406 Sulphurous Acid,.... 7036 25 460 Cyanogen, 8216 27'7 More recent, and probably more accurate observations, which will be published in vol. v. of " Travaux et Memoires du Bureau International des Poids et Mesures," 1m ve been conducted by Benoit with Fizeau's dilatometer. They give 1-0002923 as the index of refraction of air for the I) line at C. wild 760 mm.; and for the temperature coefficient they give 003667, whi.-h is identical with the coefficient of expansion of air. The larger value, -00382, obtained by Mascart, is traced to imperfect measurement of temperature. Coefficient of Dispersive Power. 97. Assuming Cauchy's formula viii.] LIGHT. 85 (where A is the wave-length), which is known to be approximately true for air within the limits of the visible spectrum, the constant b may be called the coefficient of dispersive power. Employing as the unit of length for A, the 10~* of a centimetre, Mascart ("Ann. de 1' Ecole Normale," 1877, p. 62) has obtained the following values for 6 : Coefficient of Dispersion. Air, -0058 Nitrogen, '0067 Oxygen, '0064 Hydrogen, '0043 Carbonic Oxide, '0075 Carbonic Acid, '0052 Nitrous Oxide, '0125 Cyanogen, '0100 According to Mascart, the ratio of dispersion to devia- tion for the two lines B and H is '024 for air, -032 for the ordinary ray in quartz, -038 for light crown glass, 040 for water, and '046 for the ordinary ray in Iceland- spar. Rotation of Plane of Polarization. 98. The rotation produced by 1 millim. of thickness of quartz cut perpendicular to the axis has the following values for different portions of the spectrum, according to the observations of Soret and Sarasin (" Com. Ren. 95," p. 635, 1882), the temperature of the quartz being 20 C. : Rotation. A 12-668 B 15-746 C 17'318 D 2 21-684 D 21-727 Rotation. E 27'543 F 32773 G 42 604 H... , 51-193 86 UNITS AND PHYSICAL CONSTANTS. [CHAP. vin. According to the same observers, the rotation at t C. is .equal to the rotation at C. multiplied by 1+-000179*. Units of Illuminating Power. 99. The British "Candle" is a spermaceti candle, J inch in diameter (6 to the lb.), burning 120 grains per hour. The French " Carcel " is a lamp of specified construc- tion, burning 42 grammes of pure Colza oil per hour. One "carcel" is equal to about 9J "candles." The unit adopted by the International Congress at Paris, April 1884, is a square centimetre of molten platinum at the temperature of solidification. The surface illuminated by it in photometric tests is to be normally opposite to the surface of the molten platinum. Accord- ing to the experiments of M. Violle the author of this unit, it is equal to 2 '08 carcels. It is therefore about 19| candles. 87 CHAPTER IX. HEAT. 100. THE unit of lieat is usually defined as the quantity of heat required to raise, by one degree, the temperature of unit mass of water, initially at a certain standard tem- perature. The standard temperature usually employed is C. ; but this is liable to the objection that ice may be present in water at this temperature. Hence 4 C. has been proposed as the standard temperature ; and another proposition is to employ as the unit of heat one hundredth part of the heat required to raise the unit mass of water from to 100 C. 101. According to Regnault (" Mem. Acad. Sciences," xxi. p. 729) the quantity of heat required to raise a given mass of water from to t 0. is proportional to t + -000 02* 2 + -000 000 3* 3 . . . . (1) The mean thermal capacity of a body between two stated temperatures is the quantity of heat required to raise it from the lower of these temperatures to the higher, divided by the difference of the temperatures. The mean thermal capacity of a given mass of water between 0. and t is therefore proportional to 1 + -000 02* + -000 000 3* 2 . ... (2) 88 UNITS AND PHYSICAL CONSTANTS. [CHAP. The thermal capacity of a body at a stated temperature is the limiting value of the mean thermal capacity as the range is indefinitely diminished. Hence the thermal capacity of a given mass of water at t is proportional to the differential coefficient of (1), that is to 1 + -000 04* + -000 000 9* 2 . ... (3) Hence the thermal capacities at and 4 are as 1 to 1-000174 nearly; and the thermal capacity at is to the mean thermal capacity between and 100 as 1 to 1-005. 102. If we agree to adopt the capacity of unit mass of water at a stated temperature as the unit of capacity, the unit of heat must be defined as n times the quantity of heat required to raise unit mass of water from this initial temperature through - of a degree when n is indefinitely n great. Supposing the standard temperature and the length of the degree of temperature to be fixed, the units both of heat and of thermal capacity vary directly as the unit of mass. In what follows, we adopt as the unit of heat (except where the contrary is stated) the heat required to raise a gramme of pure water through 1 C. at a temperature intermediate between and 4. This specification is sufficiently precise for the statement of any thermal measurements hitherto made. 103. The thermal capacity of unit mass of a substance at any temperature is called the specific heat of the sub- stance at that temperature ix.] HEAT. 89 Specific heat is of zero dimensions in length, mass, and time. It is in fact the ratio increment of heat in the substance increment of heat in water for a given increment of temperature, the comparison being between equal masses of the substance at the actual temperature and of water at the standard temperature. The numerical value of a given concrete specific heat merely depends upon the standard temperature at which the specific heat of water is called unity. 104. The thermal capacity of unit volume of a sub- stance is another important element : we shall denote it by c. Let s denote the specific heat, and d the density of the substance ; then c is the thermal capacity of d units of mass, and therefore c = sd. The dimensions of c in length, mass, and time are the same as those of d, namely, . Its numerical value will not be altered by any change in the units of length, mass, and time, which leaves the value of the density of water unchanged. In the O.G.S. system, since the density of water between and 4 is very approximately unity, the thermal capacity of unit volume of a substance is the value of the ratio increment of heat in the substance increment of heat in water for a given increment of temperature, when the compari- son is between equal volumes. 105. Mr. Herbert Tomlinson (" Proc. Roy. Soc.," June 19, 1885) has obtained the following determinations of specific heat from observations conducted in a uniform 90 UNITS AND PHYSICAL CONSTANTS. [CHAP. manner with metallic wires well annealed. The wires were heated sometimes to 60 C. and sometimes to 100 C., and were plunged in water at 20. The formula? are for the true specific heat at t C : Aluminium, '20700 + '0002304* Copper, -09008+ '0000648* German Silver, "09413 + '0000106* Iron, -10601 + '000 140* Lead, : '02998+ '000031* Platinum, '03198 + '000013* Platinum Silver, -04726 + '000028* Silver, -05466 + '000044? Tin, -05231 + '000072* Zinc, '09009+ '000075* The formula? for the mean specific heat between and t* are obtained from these by leaving the first term un- changed and halving the second term. Yiolle has made the following determinations of specific heat at t : Platinum, '0317+ '000012* Iridium, '0317+ '000012* Palladium, -0582 + '000020* H. F. Weber has determined the specific heat of diamond to be 0947 + -000 994* - -000 000 3G* 2 , and consequently the mean specific heat of diamond from to t to be 0947 + -000 497* - -000 000 12* 2 . The mean specific heat of ice according to Regnault is 504 between - 20 and 0, and '474 between - 78 and 0. 106. The following list of specific heats of elementary substances is condensed from that given in Landolt and Bornstein's tables : II.] HEAT. 91 Substance. Temperature. Sp. Heat. Observer. Aluminium, 15 to 97 2122 Regnault. Antimony, . 13 106 0486 Bede. Arsenic (crystalline), . 21 68 0830 ( Bettendorff j Wullner. ,, (amorphous), . 21 65 0758 " Bismuth, , 9 102 0298 Bede. Borax (crystalline), , 100 2518 Mixter&Dana , , (amorphous), 13 48 254 Kopp. Bromine, solid, -78 ,,-20 0843 Regnault. ,, liquid, 13 45 1071 Andrews. Cadmium, 100 0548 Bunsen. Calcium, 100 1804 Carbon, diamond, 11 112S H. F. Weber. graphite, 11 1604 ,, ,, wood charcoal,.. to 99 1935 > Cobalt, 9 97 1067 Regnault. Copper, 15 100 0933 Bede. Gold, 100 0316 Violle. Iodine, 9 ,, 98 0541 Regnault. Iridium, 100 0323 Violle. Iron, 50 1124 Bystrom. Lead, 19 to 48 0315 Kopp. Lithium, 27 99 9408 Regnault. Magnesium , 20 51 245 Kopp. Manganese, 14 ,, 97 1217 Regnault. Mercury, solid, -78 ,,-40 0319 ,, liquid, 17 48 0335 Kopp. Molybdenum, 5 15 0659 ( De la Rive and ( Marcet. Nickel, 14 97 1092 Regnault. Osmium, 19 98 0311 5 Palladium, 100 0592 Violle. Phosphorus ( yellow , solid ) -78 10 1699 Regnault. ( ,, liquid) 49 98 2045 Person. (red), 15 98 1698 Regnault. Platinum, ,, 100 0323 Violle. Potassium, -78 1655 Regnault. UNITS AND PHYSICAL CONSTANTS. [CHAP. Substance. Temperature. Sp. Heat. Observer. Rhodium, 10 97 0580 Regnault. Selenium, crystalline, 22 62 0840 j Bettendorff & j Wlillner. Silicon, crystalline, 22 1697 H. F. Weber. Silver, to 100 0559 Bunsen. Sodium, -28 6 2934 Regnault. Sulphur (rhomb, cryst.), 17 45 163 Kopp. ,, (newly melted), 15 97 1844 Regnault. Tellurium, crystalline, 21 51 0475 Kopp. Thallium, 17 100 0335 Regnault. Tin, cast, 100 0559 Bunsen. Zinc, 0",, 100 0935 J5 Substances not Elementary. Brass (4 copper 1 tin), hard, 15 to 98 '0858 Regnault. soft, 14 98 '0862 Ice, -20,, '504 107. The following determinations of specific heat of liquids are by Regnault. We have omitted decimal figures after the fourth, as even the second figure is different with different observers : Alcohol. Temp. Sp. Ht. -20 -5053 Chloroform. Temp. 8p. Ht. - 30 -2293 Oil of Turpentine. Temp. Sp. Ht. -20 -3842 5475 2324 4106 40 6479 30 2354 40 4538 80 7694 60 2384 80 4842 120 5019 160 5068 Ether. Temp. Sp. Ht. -30 -5113 Bisulphide of Carbon. Temp. Sp. Ht. -30 -2303 5290 2352 30 5468 30 2401 Schiiller has found the specific heat of liquid benzine at to be 37980 + -00144*. IX.] HEAT. 93 108. The following table (from Miller's "Chemical Physics," p. 308) contains the results of Regnault's ex- periments on the specific heat of gases. The column headed " equal weights " contains the specific heats in the sense in which we have defined that term. The column headed "equal volumes" gives the relative thermal capa- cities of equal volumes at the same pressure and tem- perature : Thermal Capacities of Gases and Vapours at Constant Pressure. Gas or Vapour. Equal Gas or Vapour. Equal Vols. Weights. Vols. Weights Air, - - - Oxygen, - 2375 2405 2375 2175 Hydrochloric \ Acid, - -/ 2352 1842 Nitrogen, - Hydrogen, 2368 2359 2438 3-4090 Sulphuretted \ Hydrogen, / 2S57 2432 Chlorine, - 2964 1210 Steam, 2969 4805 Bromine, - 3040 0555 Alcohol, - 7171 4584 Nitrous Oxide, 3447 2262 Wood Spirit, - 5063 4580 Nitric Oxide, - 2406 2317 Ether, - - j 1 -2-266 4796 Carbonic Oxide 2370 2450 Ethyl Chloride, 6096 2738 Carbonic ) Anhydride, / Carbonic Di- \ sulphide, / 3307 4122 2163 1569 ,, Bromide, Disul- \ phide,/ ,, Cyanide, 7026 1-2466 8293 1896 4008 4261 Ammonia, 2996 5084 Chloroform, - 0401 1566 Marsh Gas, 3277 5929 Dutch Liquid, 7911 2293 OlefiantGas, - 4106 4040 Acetic Ether, - 1 -2184 4008 Arsenious \ Chloride, J 7013 1122 Benzol, Acetone, - 1-0114 8341 :^7o4 4125 Silicic Chloride Titanic 7778 8564 1322 1290 OilofTurpen-| tine, - - J 2-3776 5061 Stannic ,, Sulphurous \ Anhydride, J 8639 341 0939 154 Phosphorus \ Chloride, / 63b6 1347 94 UNITS AND PHYSICAL CONSTANTS. [CHAP. 109. E. Wiedemann (" Pogg. Ann.," 1876, No. 1, p. 39) has made the following determinations of the specific heats of gases : Specific Heat. AtO. At 100. At 200. Jjjjg Air, 0-2389 ... ... 1 Hydrogen, 3 '410 ... ... 0*0692 Carbonic Oxide, 0-2426 ... ... 0*967 Carbonic Acid, 0*1952 0*2169 0'23S7 1-529 Ethyl, 0-3364 0-4189 0'5015 0'9677 Nitric Oxide, 0*1983 0*2212 0*2442 1*5241 Ammonia, 0*5009 0*5317 0*5629 0*5894 Multiplying the specific heat by the relative density, he obtains the following values of Thermal Capacity of Equal Volumes. At O c . At 100. At 200. Air, 0*2389 Hydrogen, 0*2359 Carbonic Oxide,.. 0*2346 Carbonic Acid,... 0*2985 0*3316 0*3650 Ethyl, 0*3254 0*4052 0*4851 Nitric Oxide, 0*3014 0*3362 0*3712 Ammonia, 0*2952 0*3134 0*3318 The same author ("Pogg. Ann.," 1877, New Series, vol. ii. p. 195) has made the following determinations of specific heats of vapours at temperature t : ^P-. $S$SBL Specific Heat. Chloroform, -J6-9 to 189*8 -1341 + '0001354* Bromic Ethyl, . . 27 *9 to 1 89 ' * 1 354 + *003560 Benzine, 34 '1 to 115 '1 '2237 + '0010228* Acetone, 26*2 to 179*3 "2984 + *0007738/ Acetic Ether, . . . 32 *9 to 1 1 3 *4 '2738 + *OOOS700 Ether, 25 *4 to 188 *8 "3725 + '0008536/ ix.j HEAT. 95 Regnault's determinations for the same vapours were as follows : Mean Specific Heat for this Range. Vanour Range of , *- . ^ Temperature. According to According to Regnault. Wiedemann. Chloroform, 117 to 228 '1567 '1573 Bromic Ethyl,... 77 7 to 196 '5 "1896 -1841 Benzine, 116 to 218 '3754 "3946 Acetone, 129 to 233 "4125 '3946 Acetic Ether,.... 115 to 219 '4008 '4190 Ether, 70 to 220 -4797 "4943 Regnault has also determined the mean specific heat of bisulphide of carbon vapour between 80 and 147 to be 1534, and between 80 and 229 to be -1613. MELTING POINTS AND HEAT OF LIQUEFACTION. 110. Violle has made the following determinations of melting points (" Com. Ren.," Ixxxix. p. 702) : Silver, 954 Gold, 1045 Copper, 1054 Palladium, 1500 Platinum, 1775 Iridium, 1950 This last temperature 1950 is very near to that of the hottest part of the oxyhydrogen flame. The same observer has found the latent heat of lique- faction of platinum to be 27 '2, and of palladium 36-3 ("Com. Ren." Ixxxv. p. 543, and Ixxxvii. p. 981). 111. The following approximate table of melting points is based on that given in the second supplement to Watt's " Dictionary of Chemistry," pp. 242, 243 : Platinum, 2000 Copper, 1090 Palladium, 1950 Silver, 1000 Gold, 1200 Borax, 1000 Cast Iron, 1200 Antimony, 432 Glass, 1100 Zinc, 360 96 UNITS AND PHYSICAL CONSTANTS. [CHAP. Lead, 330 Cadmium, 320 Bismuth, 265 Tin, 230 Selenium, 217 Cane Sugar, 160 Sulphur, Ill Sodium, .. Wax, 68 Potassium, 58 Paraffin, 54 Spermaceti, 44 Phosphorus, 43 Water, Bromine, -21 Mercury, -40 Mercury, Phosphorus, . . Lead, Melting Point. -39 44 332 Latent Heat. 2-82 5-0 5-4 Sulphur, Iodine, 115 107 9-4 117 Bismuth, Cadmium, Tin, .. 270 320 235 12-6 13-6 14-25 Sodium, 90 112. The following table (from Watt's "Dictionary of Chemistry," vol. iii. p. 77) exhibits the latent heats of fluidity of certain substances, together with their melting points : Melting Latent Point. Heat. Silver, 1000 21 '1 Zinc, 433 28'1 Chloride of Calcium (CaC1.3H 2 0),.... 28-5 40'7 Nitrate of Potas- sium, 339 47'4 Nitrate of Sodium, 310'5 63 '0 The latent heat of fluidity of water was found by Regnault, and by Provostaye and Desains, to be 79. Bunsen, by means of his ice-calorimeter (" Pogg. Ann.,' 7 vol. cxli. p. 30), has obtained the value 80-025. He finds the specific gravity of ice to be '9167. 113. Chandler Roberts and Wrightson have compared the densities of molten and solid metals by weighing a solid metal ball in a bath of molten metal either of the same or a different kind (" Phys. Soc.," 1881, p. 195, and 1882, p. 102). They find that "iron expands rapidly (as much as 6 per cent.) in cooling from the liquid to the plastic state, and then contracts 7 per cent, to solidity ; whereas bismuth appears to expand in cooling from the liquid to the solid state about 2 -35 per cent." The following is their tabular statement of results : IX.] HEAT. 97 Metal. Bismuth Sp. Grav. of Solid. 9 -82 Sp. Grav. of Liquid. 10-055 Percentage of change in volume from cold solid to liquid. Decrease of vol. 2*3 Copper, ... 8-8 8-217 Increase of vol. 7'1 Lead 11 '4 10-37 ,, 9-93 Tin, . . . 7'5 7-025 6-76 Zinc, 7-2 6-48 11-1 Silver. .. ,. 10-57 9-51 11-2 Irpn (No. 4 foundry, ) fi . q5 fi . 8g Cleveland), j b 1-02 114. Change of volume in melting, from Kopp's experi- ments (Watt's "Die.," Art. Heat, p. 78) : Phosphorus. Calling the volume at unity, the volume at the melting point (44) is 1*017 in the solid, and 1-052 in the liquid, state. Sulphur. Volume at being 1, volume at the melting point (115) is 1-096 in the solid, and 1'150 in the liquid, state. Wax. Volume at being 1, volume at melting point (64) is 1-161 in solid, and 1-166 in liquid, state. Stearic Acid. Volume at being 1, volume at melting point (70) is 1-079 in solid, and 1-198 in liquid, state. Ro*e? 8 Fusible Metal (2 parts bismuth, 1 tin, 1 lead). Volume at being 1, volume at 59 is a maximum, and is 1*0027. Volume at melting point (between 95 and 98) is greater in liquid than in solid state by 1 '55 per cent. 115. The following table (from Miller's "Chemical Physics," p. 344) exhibits the change of volume of certain substances in passing from the liquid to the vaporous condition under the pressure of one atmosphere : 1 volume of water yields 1696 volumes of vapour. ,, alcohol 528 ether 298 ,, oil of turpentine 193 ,, ,. G 98 UNITS AND PHYSICAL CONSTANTS. [CHAP. 116. The following table of boiling points and heats of vaporization, at atmospheric pressure, is condensed from Landolt and Bornstein, pp. 189, 190 : Boiling Latent Heat of observer Point. Vaporization. Observer. Alcohol, 77'9 202-4 Andrews. Bisulphide of Carbon, 46 '2 86 '7 Bromine, 58 45'6 ,, Ether, 34'9 90'4 Mercury, 350 62'0 Person. Oil of Turpentine, 159'3 74 '0 Brix. Sulphur,.. 316 362'0 Person. Water, 100 535'9 Andrews. 117. Regnault's approximate formula for what he calls "the total heat of steam at t" that is, for the heat required to raise unit mass of water from to t in the liquid state and then convert it into steam at t, is 606-5 + -3052. If the specific heat of water were the same at all tempera- tures, this would give 606-5 --695* as the heat of evaporation at t. But since, according to Regnault, the heat required to raise the water from to t is *+-00002 2 + -000 000 3* 3 , the heat of evaporation will be the difference between this and the " total heat," that is, will be 606-5 - -695* - -000 02* 2 - -000 000 3* 3 , which is accordingly the value adopted by Regnault as the heat of evaporation of water at t. 118. According to Regnault, the increase of pressure at constant volume, and increase of volume at constant ix.] HEAT. 99 pressure, when the temperature increases from to 100, have the following values for the gases named : Gas. At Constant At Constant Volume. Pressure. Hydrogen "3667 '3661 Air, -3665 '3670 Nitrogen, '3668 Carbonic Oxide, '3667 '3669 Carbonic Acid, '3688 '3710 Nitrous Oxide, '3676 '3719 Sulphurous Acid, -3845 '3903 Cyanogen, '3829 '3877 Jolly has obtained the following values for the coeffi- cient of increase of pressure at constant volume : Air, -00366957 Oxygen, -00367430 Hydrogen, '00365620 Nitrogen, '0036677 Carbonic Acid, '0037060 Nitrous Oxide, '0037067 Mendelejetf and Kaiander have determined the co- efficient of expansion of air at constant pressure to be 0036843. 119. Regnault's results as to the departures from Boyle's law are given in the form Ii = lA(m-l)B(m-l)2, v o r o Vj denoting the volume at the pressure P p V the volume TjT at atmospheric pressure P . and m the ratio . ^i For air, the negative sign is prefixed to A and the posi- tive sign to B, and we have log A = 3-0435120, log 6 = 5-2873751. 100 UNITS AND PHYSICAL CONSTANTS. [CHAP. For nitrogen, the signs are the same as for air, and we have log A = 4-8399375, log B = 6-8476020. For carbonic acid, the negative sign is to be prefixed both to A and B, and we have log A = 3-9310399, log B- 6-8624721. For hydrogen, the positive sign is to be prefixed both to A and B, and we have log A = 4-7381736, log B = 6-9250787. 120. The following table, showing the maximum pres- sure of aqueous vapour at temperatures near the ordinary boiling point, is based on Regnault's determinations, as revised by Moritz (Guyot's Tables, second edition, collec- tion D, table xxv. ) : Centims. of TV*.* >r Temperature. Mercury * 99-0 73-319 9-779 x 10 5 99-1 73-584 9'814 99-2 73-849 9 -849 ,, 99-3 74-115 9-885 ,, 99-4 74-382 9 '920 ,, 99-5 74-650 9 '956 99-6 74-918 9 "992 997 75-187 1 -0028 x 10 s 99-8 75-457 1-0064 99-9 75-728 1-0100 ,, 100-0 76-000 1-0136 100-1 76-273 1-0173 ,, 100-2 76-546 1-0209 ,, 100-3 76-820 1-0245 ,, IX.] HEAT. 101 Temperature. 100-4 100-5 100-6 100-7 100-8 100-9 101-0 Centims. of Mercury at Paris. 77-095 77-371 77-647 77-925 78-203 78-482 78-762 Dynes per sq. cm. l-0282x!0 6 1-0319 1-0356 ,, 1-0393 ,, 1-0430 1-0467 1-0505 , 121. Maximum Pressure of Aqueous Vapour at various temperatures, in dynes per sq. centim. -20 15 -10 - 5 5 10 15 1236 1866 2790 4150 6133 8710 12220 16930 20 23190 25 31400 30 42050 40 73200 50 l-226x!0 5 60 1-985 80 4-729 100 l-014xl0 6 120 1-988 140 3-626 160 6-210 180 l-006xl0 7 200 1 -560 Maximum Pressure of various Vapour s, in dynes per sq. cm. Alcohol. Ether. Sulphide of Carbon. Chloroform. -20 4455 9-19 x 10 4 6-31 x 10 4 -10 8630 1-53 x 10 5 1-058 x 10 5 16940 2-46 1-706 10 32310 3-826 2-648 20 59310 5-772 3-975 2-141 x 10 5 30 1-048 x 10 5 8-468 5-799 3-301 40 1-783 , 1-210 x 10 6 8-240 4927 50 2-932 , 1-687 1-144 x 10 6 7-14 60 4-671 , 2-301 1-554 1-007 x 10 6 80 1-084 x 10 6 4-031 2-711 1-878 100 2-265 , 6-608 4-435 3-24 120 4-31 1-029 x 10 7 6-87 5-24 102 UNITS AND PHYSICAL CONSTANTS. [CHAP. 122. The following are approximate values of the maximum pressure of aqueous vapour at various tempera- tures, in millimetres of mercury. They can be reduced to dynes per sq. cm. by multiplying by 133*4 : mm. mm. mm. mm. 4-6 92 567 112 1150 132 2155 10 9-2 94 611 114 1228 134 2286 20 17-4 96 658 116 1311 136 2423 30 31-fi 98 707 118 1399 138 2567 40 54-9 100 760 120 1491 140 2718 50 96-2 102 816 122 1588 142 2875 60 149 - 104 875 124 1691 144 3040 70 233 106 938 126 1798 146 3213 80 355 108 1004 128 1911 148 3393 90 525 110 1075 130 2030 150 3581 123. The density (in gm. per cub. cm.) of aqueous vapour at any temperature t and any pressure p (dynesper sq. cm.), whether equal to or less than the maximum, pressure, is 622 x -001276 y p 1+-00366* X 106' , If q denote the pressure in millimetres of mercury, the approximate formula is 622 x -001 293 q 1 x -00366(5 X 760* 124. Temperature of evaporation and dew-point (Glaisher's Tables, second edition, page iv.). The fol- lowing are the factors by which it is necessary to mul- tiply the excess of the reading of the dry thermometer over that of the wet, to give the excess of the tempera- ture of the air above that of the dew-point : Beading of Dry Bulb Therm. -10C.=14F. - 5 23 32 + 5 41 + 10 50 Factor. 8-76 7'28 3-32 2-26 2-06 Beading of Dry Bulb Therm. 15C.=:59 F. 20 68 25 77 30 86 35 95 Factor. 1-89 1-79 T70 1-65 1-60 IX.] HEAT. 103 125. Critical temperatures of gases, above which they cannot be liquefied (abridged from Landolt and Bornstein, p. 62) :- Critical mperature. Max. Pressure of Gas at this Temp. -174-2 98'9atm. - 105-4 48-7 ,, - 123-8 42-1 ,, 30-92 J 32-0 77-0 271-8 74-7 55 155-4 78-9 ,, 260-0 54-9 > 280-6 49-5 J 5 234-3 62-1 }1 190-0 36-9 >f Observer. Sarrau. Andrews. Sarrau. Sajotschewsky. Hydrogen, Oxygen, Nitrogen, Carbonic Acid, Bisulphide of Carbon, Sulphurous Acid, Chloroform, Benzol, Alcohol, Ether, Conductivity. 126. By the thermal conductivity of a substance at a given temperature is meant the value of k in the expression (1) where Q denotes the quantity of heat that flows, in time t, through a plate of the substance of thickness x, the area of each of the two opposite faces of the plate being A, and the temperatures of these faces being respectively Vj and -z> 2 , each differing but little from the given temper- ature. The lines of flow of heat are supposed to be normal to the faces, or, in other words, the isothermal surfaces within the plate are supposed to be parallel to the faces ; and the flow of heat is supposed to be steady, in other words, no part of the plate is to be gaining or losing heat on the whole. 104 UNITS AND PHYSICAL CONSTANTS. [CHAP. Briefly, and subject to these understandings, conduc- tivity may be denned as the quantity of heat that passes in unit time, through unit area of a plate whose thickness is unity, when its opposite faces differ in temperature by one degree. 127. Dimensions of Conductivity. From equation (1) we have = _Q (2) v. 2 - v l At The dimensions of the factor are simply M. since v 2 -v l the unit of heat varies jointly as the unit of mass and the length of the degree. The dimensions of the factor are - ; hence the dimensions of k are =_ . This is on the supposition that the unit of heat is the heat required to raise unit mass of water one degree. In calculations relating to conductivity it is perhaps more usual to adopt as the unit of heat the heat required to raise unit volume of water one degree. The dimensions of O L" will then be L 3 , and the dimensions of k will be 7= v^-v These conclusions may be otherwise expressed by say- M ing that the dimensions of conductivity are ^ when the JL1 thermal capacity of unit mass of water is taken as unity, T ^ and are when the capacity of unit volume of water is taken as unity. In the C.G.S. system the capacities of unit mass and unit volume of water are practically identical. ix.] HEAT. 105 128. Let c denote the thermal capacity of unit volume of a substance through which heat is being conducted. Then - denotes a quantity whose value it is often neces- C sary to discuss in investigations relating to the transmis- sion of heat. We have, from equation (2), c v 2 v l At Q k where Q' denotes . Hence - would be the numerical c c value of the conductivity of the substance, if the unit of heat employed were the heat required to raise unit volume of the substance one degree. Professor Clerk Maxwell k proposed to call - the tJiermometric conductivity, as dis- tinguished from k the thermal or calorimetric conductivity. We prefer, in accordance with Sir Wm. Thomson's article, k " Heat," in the Encyclopaedia Britannica, to call - the diffusivity of the substance for heat, a name which is k based on the analogy of - to a coefficient of diffusion. C Coefficient of Diffusion. 129. There is a close analogy between conduction and diffusion. Let x denote the distance between two parallel plane sections A and B to which the diffusion is perpendicular, and let these sections be maintained in constant states. Then, if we suppose one substance to be at rest, and another substance to be diffusing through it, the coefficient of diffusion K is defined by the equation y=K* ...... (i) 106 UNITS AND PHYSICAL CONSTANTS. [CHAP. where y denotes the thickness of a stratum of the mixture as it exists at B, which would be reduced to the state existing at A by the addition to it of the quantity which diffuses from A to B in the time t. When the thing diffused is heat, the states at A and B are the temperatures v 1 and v 2 , and y denotes the thickness of a stratum at the lower temperature which would be raised to the higher by the addition of as much heat as passes in the time t. This quantity of heat, for unit area, will be kt , -(%-%) JL which must therefore be equal to yc(v 2 - vj, whence we have k t y = ~ -- C X k The "thermometric conductivity" - may therefore be re- c garded as the coefficient of diffusion of heat. 130. When we are dealing with the mutual inter- diffusion of two liquids, or of two gases contained in a closed vessel, subject in both cases to the law that the volume of a mixture of the two substances is the sum of the volumes of its components at the same pressure, the quantity of one of the substances which passes any section in one direction must be equal (in volume) to the quantity of the other which passes it in the opposite direction, since the total volume on either side of the section remains unaltered ; and a similar equality must hold for the quantities which pass across the interval between two sections, provided that the absorption in the interval ix.] HEAT. 107 itself is negligible. Let x as before denote the distance between two parallel plane sections A and B to which the diffusion is perpendicular. Let the mixture at A consist of in parts by volume of the first substance to 1 - in of the second, and the mixture at B consist of n parts of the second to 1 - n of the first, m being greater than 1 - n, and therefore n greater than 1 - m. The first substance will then diffuse from A to B, and the second in equal quantity from B to A. Let each of these quantities be such as would form a stratum of thickness z (the vessel being supposed prismatic or cylindrical, and the sections considered being normal sections), then s will be propor- tional to m (\ n) m + n 1 ^t. that is to t. x x and the coefficient of interdiffusion K is defined by the equation _ 1 < ..... (2) The numerical quantity m + n - 1 may be regarded as measuring the difference of states of the two sections A and B. If y now denote the thickness of a stratum in the con- dition of B which would be reduced to the state existing at A by the abstraction of a thickness z of the second substance, and the addition of the same thickness of the first, we have (\n}y + z as the expression for the quantity of the first substance in the stratum after the operation. This is to be equal to my. Hence we have 108 UNITS AND PHYSICAL CONSTANTS. [CHAP. ^ind substituting for z its value in (2) we have finally = K|, (4) which is of the same form as equation (1), y now denoting the thickness of a stratum of the mixture as it exists at B, which would be reduced to the state existing at A by the addition to it of the quantity of one substance which diffuses from A to B in the time t, and the removal from it of the quantity of the other substance which diffuses from B to A in the same time. 131. The following values of K in terms of the centi- metre and second are given in Professor Clerk Maxwell's " Theory of Heat," 4th edition, p. 332, on the authority of Professor Loschmidt of Vienna. Coefficients of Interdiffusion of Gases. Carbonic Acid and Air, -1423 ,, ,, Hydrogen, -5614 ,, ,, Oxygen, -1409 ,, Marsh Gas, -1586 ,, ,, Carbonic Oxide, -1406 ,, ,, Nitrous Oxide, -0982 Oxygen and Hydrogen, '7214 ,, ,, Carbonic Oxide, '1802 Carbonic Oxide and Hydrogen, '6422 Sulphurous Acide and Hydrogen, '4800 k These may be compared with the value of - for air, c which, according to Professor J. Stefan of Vienna, is '256. The value of k for air, according to the same authority, is 5*58 x 10~ 5 , and is independent of the pressure. Pro- fessor Maxwell, by a different method, calculates its value at 5-4 x 10-1 ix.] HEAT. 109 Results of Experiments on Conductivity of Solids. 132. Principal Forbes' results for the conductivity of iron (Stewart on Heat, p. 261, second edition) are ex- pressed in terms of the foot and minute, the cubic foot of water being the unit of thermal capacity. Hence the value of Forbes' unit of conductivity, when referred to C.G.S., is pjr , or 15 '48; and his results must be multiplied by 15 '48 to reduce them to the C.G.S. scale. His observations were made on two square bars ; the side of the one being 1^ inch, and of the other an inch. The results when reduced to C.G.S. units are as follows : H-inch bar. 1-inch bar. ......... -207 ......... '1536 25 ......... -1912 ......... -1460 50 ........ -1771 ......... -1399 75 ......... -1656 ......... -1339 100 ......... -1567 ......... '1293 125 ......... -1496 ........ -1259 150 ......... -1446 ......... -1231 175 ......... -1399 ......... -1206 200 ......... -1356 ......... -1183 225 ......... -1317 ......... -1160 250 ......... -1279 ......... -1140 275 ......... -1240 ......... -1121 133. Neumann's results ("Ann. de. Chim." vol. Ixvi. p. 185) must be multiplied by -000848 to reduce them to our scale. They then become as follows : Copper, .............................. 1-108 Brass, ................................. -302 Zinc, ................................... -307 Iron, ................................... -164 German Silver, ..................... -109 Ice, ......................... -0057 110 UNITS AND PHYSICAL CONSTANTS. [CHAP. In the same paper he gives for the following substances k k the values of i or - ; that is, the quotient of conductivity by the thermal capacity of unit volume. These require the same reducing factor as the values of k, and when reduced to our scale are as follows : Values of 1 Coal, -00116 Melted Sulphur, '00142 Ice, -0114 Snow, -00356 Frozen Mould, -00916 Sandy Loam, -0136 Granite (coarse), '0109 Serpentine, "00594 134. Sir W. Thomson's results, deduced from observa- tions of underground thermometers at three stations at Edinburgh ("Trans. R. S. E.," 1860, p. 426), are given in terms of the foot and second, the thermal capacity of a cubic foot of water being unity, and must be multiplied by (30 -48) 2 or 929 to reduce them to our scale. The following are the reduced results : k, or k Conductivity. " c " Trap-rock of Calton Hill, -00415 '00786 Sand of experimental garden, '00262 , '00872 Sandstone of Craigleith Quarry, '01068 '0231 1 My own result for the value of from the Greenwich C underground thermometers ("Greenwich Observations," 1860) is in terms of the French foot and the year. As a French foot is 32'5 centims., and a year is 31557000 seconds the reducing factor is (32'5) 2 -j- 31557000; that is, 3-347 x 10~ 5 . The result is fc c Gravel of Greenwich Observatory Hill, '01249 ix.] HEAT. Ill Professors Ayrton and Perry (" Phil. Mag.," April, 1878) determined the conductivity of a Japanese building stone (porphyritic trachyte) to be '0059. 1 35. Angstrom, in " Pogg. Ann.," vols. cxiv. (1861) and cxviii. (1863), employs as units the centimetre and the minute ; hence his results must be divided by 60. These results, as given at p. 294 of his second paper, will then stand as follows : Value of -'. Copper, first specimen, ........ 1 "216 (1 - "00214 t) ,, second specimen, ...... 1-163 (1 - '001519 t] Iron, ................................. -224 (1 - '002874 t) He adopts for c the values 84476 for copper ; '88620 for iron, and thus deduces the following values of k : Conductivity. Copper, first specimen, ......... T027 (1 - '00214 t) second specimen, ...... '983 (1 - '001519 t) Iron, .................................. -199 (1 - '002874 t) 136. A Committee, consisting of Professors Herschel and Lebour, and Mr. J. F. Dunn, appointed by the British Association to determine the thermal conductivities of certain rocks, have obtained results from which the following selection has been made by Professor Herschel : Substance. Iron pyrites, more than ........... '01 more than '0170 Rock salt, rough crystal, .......... '0113 '0288 Fluorspar, rough crystal, ......... '00963 '0156 Quartz, opaque crystal, and quartzites, ......................... -0080 to '0092 '0175 to '0190 Silicious sandstones (slightly wet), '00641 to '00854 '0130 to '0230 112 UNITS AND PHYSICAL CONSTANTS. [CHAP. Suhstanrp Conductivity in k C.G.S. Units. ~ c - Galena, rough crystal, inter- spersed with quartz, "00705 '0171 Sandstone and hard grit, dry, ... -00545 to '00565 '0120 Sandstone and hard grit, thor- oughly wet, -00590 to -00610 -0100 Micaceous flagstone, along the cleavage, -00632 '0116 Micaceous flagstone, across cleav- age, -00441 -0087 Slate, along cleavage, '00550 to '00650 '0102 Do. , across cleavage, -003 1 5 to '00360 '0057 Granite, various specimens, about '00510 to '00550 '0100 to '0120 Marbles, limestone, calcite, and compact dolomite, '00470 to '00560 '0085 to '0095 Red Serpentine (Cornwall), '00441 '0065 Caen stone (building limestone), -00433 '0089 Whinstone, trap rock, and mica schist, '00280 to '00480 '0055 to '0095 Clay slate (Devonshire), '00272 '0053 Tough clay (sun-dried), '00223 '0048 Do., soft (with one-fourth of its weight of water), 00310 '0035 Chalk, '00200 to '00330 '0046 to '0059 Calcareous sandstone (firestone), '00211 '0049 Plate-glass German and English, '00198 to '00234 '00395 to ? German glass toughened, '00185 '00395 Heavy spar, opaque rough crystal, '00177 Fire-brick, '00174 '0053 Fine red brick, '00147 '0044 Fine plaster of Paris, dry plate, '00120 ' 006 \about Do., thoroughly wet, '00160 '0025 / White sand, dry, '00093 '0032 Do. , saturated with water, about '00700 '0120 about House coal and cannel coal, '00057 to -00113 '0012 to '002T Pumice stone, '00055 ix.] HEAT. 113 137. Peclet in " Annales de Chimie," ser. 4, torn. ii. p. 114 [1841], employs as the unit of conductivity the trans- mission, in one second, through a plate a metre square and a millimetre thick, of as much heat as will raise a cubic decimetre (strictly a kilogramme) of water one degree. Formula (2) shows that the value of this conduc- tivity in the C.G.S. system, is 1000 T \j ...-. 1 ^ liooo > thatls ' Too' His results must accordingly be divided by 100. The same author published in 1853 a greatly extended series of observations, in a work entitled "Nouveaux documents relatifs aux chauffage et a la ventilation." In this series the conductivity which is adopted as unity is the transmission, in one hour, through a plate a metre square and a metre thick, of as much heat as will raise a kilogramme of water one degree. This conductivity, in C.G.S. units is 1000 100 _J_ m , . 1 1 '10000 '3600' S '360' The results must therefore be divided by 360. Those of them which refer to metals appear to be much too small. The following are for badly conducting substances : Density. Conductivity. Fir, across fibres, -48 '00026 ,, along fibres, -48 '00047 Walnut, across fibres, '00029 ,, alongfibres, '00048 Oak, across fibres, '00059 Cork, -22 '00029 Caoutchouc, '00041 Guttapercha, '00048 Starchpaste, 1-017 '00118 Glass, 2'44 '0021 H 114 UNITS AND PHYSICAL CONSTANTS. [CHAP. Density. Conductivity. Glass, 2-55 '0024 Sand, quartz, 1-47 "00075 Brick, pounded, coarse-grained, . 1 '0 "00039 passed through { , . silk sieve,.... I 1 ' 6 Fine brick dust, obtained by decan- \ , .__ tation, J L Chalk, powdered, slightly damp, '92 '00030 ,, washed and dried, "85 '00024 washed, dried, and \ , . ft9 , nft9Q compressed, J 1 Potato-starch, '71 '00027 Wood-ashes, -45 '00018 Mahogany sawdust, '31 '00018 Wood charcoal, ordinary,powdered, '49 '00022 Bakers' breeze, in powder, passed ^ . - throughsilk sieve, .../ 2o Ordinary wood charcoal in powder, ) ,, -nnnoor passed through silk sieve, \ ' Coke, powdered, '77 "00044 Iron filings, 2"05 '00044 Binoxide of man ganese, 1 '46 "00045 Woolly Substances. Cotton Wool, of all densities, "0001 1 1 Cotton swansdown (molleton de ) .nnm 1 1 coton), of all densities, / "' Calico, new, of all densities, "000139 Wool, carded, of all densities, "000122 Woollen swansdown (molleton de \ nnnnfi'7 laine) of all densities, / "" Eider-down, "000108 . Hempen cloth, new, "54 -000144 old, "58 "000119 Writing-paper, white, '85 "0001 19 Grey paper, unsized, "48 '000094 138. In Professor George Forbes's paper on conductivity (" Proc. K. S. E.," February, 1873) the units are the centim. the minute ; hence his results must be divided by 60. IX.] HEAT. 115 In a letter dated March 4, 1884, to the author of this work, Professor Forbes remarks that the mean tempera- ture of the substances in these experiments was - 10, and expresses the opinion that bad conductors (such as most of these substances) conduct worse at low than at high temperatures an opinion which was suggested by the analogy of electrical insulators. His results reduced to C.G.S. are- Ice, along axis, '00223 Ice, perpendicular to \ nfto1 o axis, J ' UUZ1 ^ Black marble, '00177 White marble, "00115 Slate, '00081 Snow, -00072 Cork, -000717 Glass, -0005 Pasteboard, -000453 Carbon, -000405 Roofing-felt, "000335 Fir, parallel to fibre, -0003 Fir, across fibre, and) along radius, / Boiler-cement, Paraffin, Sand, very fine, 00011 000 089 000 088 000162 00014 000131 Kamptulikon, Vulcanized india- rubber, Horn, ' '000087 Beeswax, 'OU0087 Felt, -000087 Vulcanite, '0000833 Haircloth, "0000402 Cotton-wool,divided, '000 0433 pressed, '0000335 Flannel, '0000355 Coarse linen, '000 0298 Quartz, along axis, '000922 00124 00057 "00083 Quartz, perpendicular) nn/in to axis, /' UU4tJ 0044 Sawdust, "000123 Professor Forbes quotes a paper by M. Lucien De la Rive ("Soc. de Ph. et d'Hist. Nat. de Geneve," 1864) in which the following result is obtained for ice, Ice, -00230. M. De la Rive's experiments are described in " Annales de Chimie," ser. 4, torn. i. pp. 504-6. 139. Dr. Robert Weber ("Bulletin, Soc. Sciences Nat. de Neufchatel," 188), has found the following conductivi- ties and surface emissivities for five specimens of rock from the St. Gothard tunnel : 116 UNITS AND PHYSICAL CONSTANTS. [CHAP. SPECIMEN No. 168. Micaceous Gneiss. Conductivity, '000917 + '0000044? Emissivity, '000185 + '0000023? Specific Heat, '1778 4- '00042? SPECIMEN No. 114. Mica Schist. Conductivity, '000733 + '000010? Emissivity, '000207 + '0000016? Specific Heat, '18000 + '00044? SPECIMEN No. 124. Eurite. Conductivity, '000862 + '00016)5 Emissivity, '000249 + "000 000 09* Specific Heat, '1682 + "0006* SPECIMEN No. 140. Gneiss. Conductivity, '0014 + '000 003? Emissivity, '00026 + '0000008? Specific Heat, '1463 +'0009? SPECIMEN No. 146. Micaceous Schist. Conductivity, '000952 + '000009? Emissivity, '000168 + '0000023? Specific Heat, '1697 +'0006? Conductivity of Liquids. 140. The conductivity of water, according to experi- ments by Mr. J. T. Bottomley ("Phil. Trans." 1881, April 3), is '002, which is nearly the same as the con- ductivity of ice. (See 138.) 141. H. F. Weber ("Sitz. kon. Preuss. Akad." 1885), has made the following determinations of conductivities of liquids at temperatures of from 9 to 15 C. He em- ploys the centimetre, the gramme, and the minute as units : we have accordingly divided the original numbers by 60 to reduce to C.G.S. IX.] HEAT. 117 Water, Conduc- tivity. 00136 000408 000670 000303 000495 000423 000373 000340 000328 000648 000472 000390 000360 000340 000325 000312 000298 000385 000378 000348 000357 000327 000335 000318 000315 000307 Amyl Acetate, Chloro Benzol, . . Conduc- tivity. 000302 000302 000288 000252 000283 000278 000284 000265 000247 000257 000278 000237 000222 000220 000208 000203 000333 000307 000272 000260 000765 000343 000382 000328 Aniline, Glycerine Ether, Chloroform, Chloro Carbon, Methyl Alcohol, Ethyl Alcohol, Propyl Alcohol, Butyl Alcohol, Amyl Alcohol, Ameisen Acid, Acetic Acid, Propion Acid, Butyric Acid, Isobutyric Acid, Valerian Acid, Isovalerian Acid, Propyl Chloride, Isobutyl Chloride, .... Amyl Chloride, . . Bromo Benzol, Ethyl Bromide, Propyl Bromide, Isobutyl Bromide,.... Amyl Bromide Ethyl Iodide, Propyl Iodide, Isobutyl Iodide, Amyl Iodide, Isocaproii Acid, Methyl Acetate, Ethyl Formiate, Ethyl Acetate, Benzol, . . Toluol, Cymol, Propyl Formiate, Propyl Acetate, Methyl Butyrate, Ethyl Butyrate, Methyl Valerate, Ethyl Valerate.... Oil of Turpentine, .... Sulphuric Acid, Bisulphide of Carbon, Oil of Mustard, Ethvl Suluhide. . . In the original paper these numbers are compared with the thermal capacities of the liquids per unit of volume, and with the calculated mean distances between their molecules. It is found that conductivity, multiplied by mean distance, divided by capacity, is a nearly constant quantity for the members of any one of the above groups. Comparing one group with another, its most widely dif- ferent values are represented by 19 and 23, if we except the last group, for which its value is between 26 and 27. Emission and Surface Conduction. 142. Mr. D. M'Farlane has published (" Proc. Roy. Soc." 118 UNITS AND PHYSICAL CONSTANTS. [CHAP. 1872, p. 93) the results of experiments on the loss of heat from blackened and polished copper in air at atmospheric pressure. They need no reduction, the units employed being the centimetre, gramme, and second. The general result is expressed by the formulas x = -000238 + 3-06 x lO' 6 * - 2-6 x 1Q-V for a blackened sui-face, and x = -000168 + 1 -98 x 10-^-1-7 x !Q- for polished copper, x denoting the quantity of heat lost per second per square centim. of surface of the copper, per degree of difference between its temperature and that of the walls of the enclosure. These latter were blackened internally, and were kept at a nearly constant temperature of 14 C. The air within the enclosure was kept moist by a saucer of water. The greatest difference of tempera- ture employed in the experiments (in other words, the highest value of t) was 50 or 60 C. The following table contains the values of x calculated from the above formula?, for every fifth degree, within the limits of the experiments : Difference of Temperature. Value of x. Ratio. Polished Surface. Blackened Surface. 5 000178 000252 707 10 000186 000266 '699 15 000193 000279 '692 20 000201 000289 '695 25 000207 000298 '694 30 000212 000306 '693 35 000217 090313 -693 40 000220 000319 '693 45 000223 000323 690 50 000225 000326 690 55 000226 000328 690 60 -000226 000328 '690 IX.] HEAT. 119 143. Professor Tait has published (" Proc. R. S. E." 1869-70, p. 207) observations by Mr. J. P. Nichol on the loss of heat from blackened and polished copper, in air, at three different pressures, the enclosure being blackened internally and surrounded by water at a temperature of approximately 8 C.* Professor Tait's units are the grain- degree for heat, the square inch for area, and the hour for time. The rate of loss per unit of area is heat emitted area x time The grain-degree is '0648 gramme-degree. The square inch is 6*4516 square centims. The hour is 3600 seconds. Hence Professor Tait's unit rate of emission is *0648 n.rrn 1 /->-<{ 6-4516 x 3600 of our units. Employing this reducing factor, Professor Tait's Table of Results will stand as follows : Pressure 1 '014 x 10 6 [760 millims. of mercury]. Bright. Temp. Cent. Loss per sq. cm. per second. 63-8 -00987 57-1 -00862 50-5 -00736 44-8 -00628 40-5 -00562 34-2 -00438 29-6 -00378 23-3 -00278 18-6 . -00210 Blackened. Temp. Cent. Less per sq. cm. per second. 61-2 -01746 50-2 -01360 41-6 '01078 34-4 "00860 27-3 -00640 20-5 -00455 *This temperature is not stated in the " Proceedings," but has been communicated to me by Professor Tait. 120 UNITS AND PHYSICAL CONSTANTS. [CHAP. Pressure 1'36-x 10 5 [102 millims. of mercury]. 62-5 -01298 I 67-8 '00492 57'5 -01158 61-1 -00433 53-2 -01048 55 -00383 47-5 -00898 43 -00791 28-5 . -00490 497 '00340 44-9 -00302 40-8 . -00268 Pressure P33 x 10 4 [10 millims. of mercury]. 62-5 01182 65 00388 57'5 54-2 ... -01074 ... -01003 60 50 -00355 . ... -00286 41-7 . . . -00726 40 00219 37-5 34 .. -00639 00569 30 23-5 -00157 00124 27-5 ... -00446 24-2 . 00391 Mechanical Equivalent of Heat. 144. The value originally deduced by Joule from his experiments on the stirring of water was 772 foot-pounds of work (at Manchester) for as much heat as raises a pound of water through 1 Fahr. This is 1389-6 foot- pounds for a pound of water raised 1 C., or 1389-6 foot- grammes for a gramme of water raised 1 C. As a foot is 30 '48 centims., and the value of g at Manchester is 981-3, this is 1389-6 x 30-48x981-3 ergs per gramme- degree; that is, 4-156 x 10 7 ergs per gramme- degree. A later determination by Joule (" Brit. Assoc. Report," 1867, pt. i. p. 522, or "Reprint of Reports on Electrical Standards," p. 186) is 25187 foot-grain-seconcl units of work per grain-degree Fahr. This is 45337 of the same units per grain-degree Centigrade, or 45337 foot-gramme- second units of work per gramme-degree Centigrade; that is to say, 45337 x (30-48) 2 = 4-212 x 10 7 ergs per gramme-degree Centigrade. ix.] HEAT. 121 In view of the fact that the B. A. standard of electrical resistance employed in this determination is now known to be too small by about 1/3 per cent., and that the cur- rent energy converted into heat was accordingly under- estimated to this extent, the result ought now to be in- creased by 1'3 per cent., which will make it 4-267 x 107. At the meeting of the Royal Society, January, 1878 (" Proceedings," vol. xxvii. p. 38), an account was given by Joule of experiments recently made by him with a view to increase the accuracy of the results given in his former paper. ("Phil. Trans.," 1850.) His latest result from the thermal effects of the friction of water, as announced at this meeting, is, that taking the unit of heat as that which can raise a pound of water, weighed in vacuo, from 60 to 61 of the mercurial Fahrenheit thermometer; its mechanical equivalent, reduced to the sea-level at the latitude of Greenwich, is 772-55 foot-pounds. To reduce this to water at C. we have to multiply by 1-00089,* giving 773-24 ft. Ibs., and to reduce to ergs per gramme-degree Centigrade we have to multiply by 981-17 x 30-48 x-. 5 The product is 4-1624 x 10 7 . 145. Some of the best determinations by various experi- menters are given (in gravitation measure) in the following list, extracted from "Watts' Dictionary of Chemistry," Supplement 1872, p. 687. The value 429 -3 in this list corresponds to 4*214 x 10 7 ergs : * This factor is found by giving t the value 15 '8 (since the tem- perature GO'S Fahr. is 15 '8 Cent.) in formula (3) of art. 101. 122 UNITS AND PHYSICAL CONSTANTS. [CHAP. Violle,... Hirn, 432 Friction of water and brass. ,, 433 Escape of water under pressure. ..... 441-6 Specific heats of air. ,, 425-2 Crushing of lead. Joule 429-3 ^ Heat produced by an electric ( current. 435 -2 (copper).. . 434*9 (aluminium) 435-8 (tin) 437-4 (lead) Regnault, 437 We shall adopt 4'2 x 10 7 ergs as the equivalent of 1 gramme-degree ; that is, employing J as usual to denote Joule's equivalent, we have J = 4-2 x 10 7 = 42 millions. 146. Heat and Energy of Combination with Oxygen. Heat produced by induced cur- rents. Velocity of sound. 1 gramme of Compound formed. Gramme- degrees of heat produced. Equivalent Energy, in ergs. Hydrogen, Carbon, H 2 CO 2 34000 A F 8000 A F l-43x!0 12 3'36x 10 11 Sulphur, . SO 2 2300 A F 9'66x 10 10 Phosphorus, Zinc, P 2 5 ZnO 5747 A 1301 A 2-41 x 10 11 5-46x 10 10 Iron, Fe 3 4 1576 A 6'62x 10 10 Tin,. SnO' 1233 A 5-18 Copper, CuO 602 A 2-53 Carbonic oxide,.... ]VIarsh-gas CO 2 CO 2 and H' J 2420 A 13100 A F 1 -02 x 10 11 5'50 Olefiant gas, . . 11900 A F 5-00 Alcohol, 6900 A F 2-90 " Combustion in Chlorine. Hydrogen, ; HC1 Potassium, KC1 Zinc, ZnCP i 23000 F T 2655 A 1529 A 9-66xlO n 1-12 6-42xl0 10 Iron Fe 2 Cl fi 1745 A 7-33 Tin, . . SnCl 4 1079 A 4-53 Copper CuCl 2 961 A 4'04 ivl 1 1 MAT. 123 The numbers in the last column are the products of the numbers in the preceding column by 42 millions. The authorities for these determinations are indicated by the initial letters A (Andrews), F (Favre and Silber- mann), T (Thomsen). Where two initial letters are given, the number adopted is intermediate between those obtained by the two experimenters. 147. Difference between the two specific heats of a gas. Let s 1 denote the specific heat of a given gas at con- stant pressure, s 2 the specific heat at constant volume, a the coefficient of expansion per degree. v the volume of 1 gramme of the gas in cubic centim. at pressure/* dynes per square centim. When a gramme of the gas is raised from to 1 at the constant pressure^;, the heat taken in is s v the increase of volume is av, and the work done against external resist- ance is avp (ergs). This work is the equivalent of the difference between s 1 and s 2 ; that is, we have s x ,s- 2 = _ JL, where J = 4'2 x 20 7 . J For dry air at the value of vp is 7*838 x 10 8 , and a is -003665. Hence we find s l - s 2 = -0684. The value of s 15 according to Eegnault, is -23 7f>. Hence the value of s z is -1691. The value of 1 ~ S -, or ^ . for dry air at and a v J megadyne per square centim. is *i -*2-' 0684 " v 7S3-8 1 24 UNITS AND PHYSICAL CONSTANTS. [CHAP. and this is also the value of - 1 2 for any other gas (at the same temperature and pressure) which has the same coefficient of expansion. 148. Change of freezing point due to change of pressure. Let the volume of the substance in the liquid state be to its volume in the solid state of 1 to 1 + e. When unit volume in the liquid state solidifies under pressure P + p, the work done by the substance is the product of P +2> by the increase of volume e } and is there- fore ~Pe +pe. If it afterwards liquefies under pressure P, the work done against the resistance of the substance is Pe ; and if the pressure be now increased to P + p, the substance will be in the same state as at first. Let T be the freezing temperature at pressure P, T + t the freezing temperature at pressure P +p, I the latent heat of liquefaction, d the density of the liquid. Then d is the mass of the substance, and Id is the heat taken in at the temperature of melting T. Hence, by thermodynamic principles, the heat converted into mechani- cal effect in the cycle of operations is T + 273 ' But the mechanical effect is pe. Hence we have t 77 pe p 3 Id ix.] HEAT. 125 - - is the lowering of the freezing-point for an additional pressure of a dyne per square centim.; and x 10 will be the lowering of the freezing point for each addi- tional atmosphere of 10 6 dynes per square centim. For water we have e = 'OS7, Z= 79-25, T = 0, d = l, Formula (3) shows that is opposite in sign to e. Hence the freezing point will be raised by pressure if the substance contracts in solidifying. 149. Change of temperature produced by adiabatic com- pression of a fluid ; that is, by compression under such circumstances that no heat enters or leaves the fluid. Let a cubic centim. of fluid at the initial temperature t C. and pressure p dynes per square centim. be put through the cycle of operations represented by the annexed "indicator diagram," ABCD, where horizontal distance from left to right denotes increase of volume and perpen- dicular distance upwards increase of pressure. In AD let the pressure be constant and equal to p. In BC let the pressure be constant and equal to p + TT, TT being small. Let AB and CD be adiabatics, so near together that AD and BC are very small compared with the altitude of the figure which is TT. 126 UNITS AND PHYSICAL CONSTANTS. [CHAP. The figure will be ultimately a parallelogram, so that the changes of volume AD and BC will be equal ; let their common value be called edt, e denoting the expansion per degree at constant pressure; dt will therefore be the difference of temperature between A and D, or between B and C. We suppose this difference to be very small compared with the difference of temperature between A and B or between C and D. The cycle is reversible; let it be performed in the direc- tion ABCD. Then heat is taken in as the substance expands from B to C, and given out as it contracts from Dto A. The work done by the substance in the cycle is equal to the area of the parallelogram, which, being the product of the base edt by the height TT, is Tredt. The heat given out in DA is Cdt, C denoting the thermal capacity of a cubic centim. of the substance at constant pressure; hence the " efficiency " is JT , and this, by the rules of JC Thermodynamics, must be equal to -j - , where r de- i i o ~\~ t notes the increase of temperature from A to B. Put T for the absolute temperature 273 + 2, then we have where r is the increase of temperature produced by the increase TT of pressure. 150. Resilience as affected by heat of compression. The expansion due to the increase of temperature r, above calculated, is re ; that is, - ~~ ; and the ratio of J C ix.] HEAT. 127 this expansion to the contraction , which would be pro- Jfi duced at constant temperature (E denoting the resilience of volume at constant temperature), is =~- : 1. Putting JO m for - , the resilience for adiabatic compression will be JC TT ; or, if m is small, E (1 + m) \ and this value is to l-m be used instead of E in calculating the change of volume due to sudden compression. The same formula expresses the value of Young's modulus of resilience, for sudden extension or compression of a solid in one direction, E now denoting the value of the modulus at constant temperature. Examples. For compression of water between 10 and 11 we have E = 2-1 x 10 10 , T = 283, e = -000 092, C - 1 ; hence For longitudinal extension of iron at 10 we have E = l-96xl0 12 , T-283, e=-0000122, C = '109x7 hence = -00234. JG Thus the heat of compression increases the volume- resilience of water at this temperature by about -| per cent., and the longitudinal resilience of iron by about J per cent. 128 UNITS AND PHYSICAL CONSTANTS. [CHAP. For dry air at and a megadyne per square centim., we have E = 10 6 , T = 273, e - _* C - -2375 x -001276, 27o = 1-404. 1 -m 151. Expansions of Volumes per degree Cent, (abridged from Watts' " Dictionary of Chemistry" Article Heat, pp. 67, 68, 71). Glass, ........................... -00002 to "000 03 Iron .............................. -000035 '000044 Copper, ......................... -000052 ,, -000057 Platinum, ..................... '000026 ,, '000029 Lead, ............................ -000084 -000089 Tin, ............................. -000058 '000069 Zinc, ............ . ................ -000087 '000090 Gold, ............................ -000044 ,, -000047 Brass, ........................... -000053 -000056 Silver, ........................ -000057 '000064 Steel, ............................ -000032 -000042 Cast Iron, ............... about '000033 These results are partly from direct observation, and partly calculated from observed linear expansion. Expansion of Mercury, according to Regnault (Watts' "Dictionary" p. 56). Te mP . = , Volume at, 6 ............... 1 '000000 ............... '00017905 10 ............... 1-001792 .............. -00017950 20 ............... 1-003590 ............... -00018001 30 ............... 1-005393 ............... -00018051 50 ......... ...... 1-009013 ............... -00018152 70 ............... 1-012655 ............... -00018253 100 ............... 1-018153 ............... -00018405 The temperatures are by air-thermometer. ix.] , HEAT. 129 The formula adopted by the Bureau International dcx Poids et Mesures for the volume at t C. (derived from Regnault's results) is 1 + -000181792*+ -000 000 000 175* 2 + 000000000035116^. Expansion of Alcohol and Ether, according to Kopp (Watts' "Dictionary," p. 62). Volume- Temp. Alcohol. Ether. 6 roooo roooo 10 1-0105 1-0152 20 1-0213 1-0312 30 1-0324 1-0483 40 . 1-0440 , 1-0667 152. Collected Data for Dry Air. Expansion from to 100 at const, pressure, as 1 to 1*367 or as 273 to 373 Specific heat at constant pressure, '2375 ,, at constant volume, -1691 Pressure-height at C., about 7 '99 x 10 5 cm., or about 26210 ft. Standard barometric column, 76 cm. = 29 '922 inches. Standard pressure, 1033 '3 gm. per sq. cm. or 14*7 Ibs. per sq. inch. or 2117 Ibs. ,, foot. or 1 '0136 x 10 6 dynes per sq. cm. Standard density, at C., "001293 gm. per cub. cm. or -0807 Ibs. per cub. foot. Standard bulkiness, 773 '3 cub . cm. per gm. or 12-39 cub. ft. per Ib. I 130 UNITS AND PHYSICAL CONSTANTS. [CHAP. ix. Dry and Moist Ait'. Mass of 1 Cubic Metre in Grammes. Temp. C. () Dry Air. 1293-1 Saturated Air. 1290*2 Vapour at Saturation. 4-9 10 ... 1247-3 . . 1241-7 9-4 20 30 . ... 1204-6 .... 1164-8 1194-3 1146*8 17-1 . ... 30-0 40 . 1127-6 1097-2 50-7 If A denote the density of dry air and W that of vapour at ituration, the den: exactly A - -608 W. o saturation, the density of saturated air is A - - W, or more 5 131 CHAPTER X. MAGNETISM. 153. THE unit magnetic pole, or the pole of unit strength, is that which repels an equal pole at unit distance with unit force. In the C.G.S. system it is the pole which repels an equal pole, at the distance of 1 centimetre, with a force of 1 dyne. If P denote the strength of a pole, it will repel an equal P 2 pole at the distance L with the force . Hence we have the dimensional equations P 2 L~ 2 = force - MLT~ 2 , P 2 = ML 3 T' 2 , P = M^TT 1 ; that is, the dimensions of a pole (or the dimensions of strength of pole) are M^L^T' 1 . 154. The work required to move a pole P from one point to another is the product of P by the difference of the magnetic potentials of the two points. Hence the dimensions of magnetic potential are >- - ML 2 T- 2 . M ~ *L ~ T 155. The intensity of a magnetic field is the force which a unit pole will experience when placed in it. Denoting 132 UNITS AND PHYSICAL CONSTANTS. [CHAP. this intensity by I, the force on a pole P will be IP. Hence IP = force = MLT- J , 1 = MLT' 3 . MLT that is, the dimensions of 'field-intensity are M*L"*T~ 1 . 156. The moment of a magnet is tbe product of the strength of either of its poles by the distance between them. Its dimensions are therefore LP; that is, M^UT' 1 . Or, more rigorously, the moment of a magnet is a quantity which, when multiplied by the intensity of a uniform field, gives the couple which the magnet ex- periences when held with its axis perpendicular to the lines of force in this field. It is therefore the quotient of a couple ML 2 T~ 2 by a field-intensity M^L'^T" 1 ; that is, it is M^L'T' 1 as before. 157. If different portions be cut from a uniformly mag- netized substance, their moments will be simply as their volumes. Hence the intensity of magnetization of a uni- formly magnetized body is defined as the quotient of its moment by its volume. But we have moment volume _ 1 ^ L _ 3 = M 4 L -4 T -i p Hence intensity of magnetization has tbe same dimensions as intensity of 'field. When a magnetic substance (whether paramagnetic or diamagnetic) is placed in a magnetic field, it is magnetized by induction, and the ratio of the intensity of the magnetization thus produced to the intensity of the field is called the " coefficient of magnetic x.l MAGNETISM. 133 induction," or " coefficient of induced magnetization," or the "magnetic susceptibility" of the substance. For paramagnetic substances (such as iron, nickel, and cobalt) this coefficient is positive ; for diamagnetic substances (such as bismuth), it is negative ; that is to say, the induced polarity is reversed, end for end, as compared with that of a paramagnetic substance placed in the same field. 158. It has generally been stated that "magnetic sus- ceptibilty " is nearly independent of the intensity of the h'eld so long as this intensity is much less than is required for saturation. But R. Shida found ("Proc. Roy. Soc.," Nov., 1882), in the softest iron wire, a, very rapid varia- tion of susceptibility at low intensities. Under the influence of the earth's vertical force at Glasgow, *545, the susceptibility had the very large value 734 when the wire was stretched by a weight, and 335 when the weight was off. Under a magnetizing force 2 -35, the susceptibilities, with and without the weight, were 235 and 154. Saturation was obtained with a magnetizing force of 80*7, which produced magnetizations 1390 and 1430, the susceptibilities being therefore 17*1 and 17 '6. With pianoforte wire (steel), the susceptibilities were 67'5 and 69'3 under the earth's vertical force, and 13*2 when saturation was just attained, with a magnetizing force of 107 '5. The magnetization at saturation was 1420, being about the same as for soft iron wire. With a square bar of soft iron nearly 1 centim. square, the susceptibility diminished from 19, under a magnetizing force of 18 -2, to 7'6, under a magnetizing force of 189, which just produced saturation. 134 UNITS AND PHYSICAL CONSTANTS. [CHAP. Examples. 1. To find the multiplier for reducing magnetic in- tensities from the foot-grain-second system to the C.G.S. system. The dimensions of the unit of intensity are In the present case we have M = -0648, L = 30'48, T = 1, since a grain is -0648 gramme, and a foot is 30 '48 centini. Hence M^L^T' 1 = ./^|= '04611 ; that is, the foot- \ olr4o grain-second unit of intensity is denoted by the number 0461 1 in the C.G.S. system. This number is accordingly the required multiplier. 2. To find the multiplier for reducing intensities from the millimetre-milligramme-second system to the C.G.S. system, we have M - T, T - 1 ~M^T,~^T~ l - / - " 1000' "16' ~V 1000" 10' Hence - - is the required multiplier. 3. Gauss states (Taylor's " Scientific Memoirs," vol. ii. p. 225) that the magnetic moment of a steel bar-magnet, of one pound weight, was found by him to be 100S77000 millimetre-milligramme-second units. Find its moment in C.G.S. units. Here the value of the unit moment employed is, in terms of C.G.S. units, M^T' 1 , where M is 10~ 3 , L is 10- 1 , arid T is 1 ; that is, its value is 10~*. 10~ = 10~ 4 . Hence the moment of the bar is 10087*7 C.G.S. units. x.] MAGNETISM. 135 4. Find the mean intensity of magnetization of the bar, assuming its specific gravity to be 7*85, and assuming that the pound mentioned in the question is the pound avoir- dupois of 453*6 grammes. Its mass in grammes, divided by its density, will be its volume in cubic centimetres \ hence we have 453-6 In! K ' 7-85 = 5778 = volume of bar. .. ,. moment 10088 ,, ntensity of magnetization = = = 174-0. volume 57-78 5. Kohlrausch states (" Physical Measurements," p. 195, English edition) that the maximum of permanent mag- netism which very thin rods can retain is about 1000 millimetre-milligramme-second units of moment for each milligramme of steel. Find the corresponding moment per gramme in C.G-.S. units, and the corresponding in- tensity of magnetization. For the moment of a milligramme we have 1000 x 10-* =10-!. For the volume of a milligramme we have (7-85)" 1 x 10~ 3 , taking 7 '85 as the density of steel. Hence the moment per gramme is 10" 1 x 10 3 = 100, and the intensity of magnetization is 100 x 7 "85 = 785. 6. The maximum intensity of magnetization for speci- mens of iron, steel, nickel, and cobalt has been deter- mined by Professor Rowland ("Phil. Mag.," 1873, vol. xlvi. p. 157, and November, 1874) that is to say, the limit to which their intensities of magnetization would approach, if they were employed as the cores of electro- magnets, and the strength of current and number of con- volutions of the coil were indefinitely increased. Professor 136 UNITS AND PHYSICAL CONSTANTS. [CHAP. Rowland's fundamental units are the metre, gramme, and second ; hence his unit of intensity is of the C.G.S. unit. His values, reduced to C.G.S. units, are At 12C. At 220C. Iron and Steel, ......................... 1390 1360 Nickel, .................................... 494 380 Cobalt, .................................... 800 (?) 7. Gauss states (loc. cit.) that the magnetic moment of the earth, in millimetre-milligramme-second measure, is 3-3092 R 3 , R denoting the earth's radius in millimetres. Reduce this value to C.G.S. Since R 3 is of the dimensions of volume, the other factor, 3 '309 2, must be of the dimensions of intensity. Hence, employing the reducing factor 10" 1 above found, we have '33092 as the corresponding factor for C.G.S. measure \ and the moment of the earth will be 33092 R 3 , R denoting the earth's radius in centimetres that is 6-37 x 10 8 . We have 33092 x (6-37 x 10 8 ) 3 - 8-55 x 10 25 for the eartlis magnetic moment in C.G.S. units. 8. From the above result, deduce the intensity of mag- netization of the earth regarded as a uniformly magnetized body. We have . , ., moment 8*55 x 10 25 n70r . intensity = _ ^7^7 = ' 790 ' volume 1-083 x 2 This is about - --- of the intensity of magnetization of .\.] MAGNETISM. 137 Gauss's pound magnet ; so that 2*2 cubic decimetres of earth would be equivalent to 1 cubic centiin. of strongly magnetized steel, if the observed effects of terrestrial mag- netism were due to uniform magnetization of the earth's substance. 9. Gauss, in his papers on terrestrial magnetism, em- ploys two different units of intensity, and makes mention of a third as " the unit in common use." The relation between them is pointed out in the passage above referred to. The total intensity at Gottingen, for the 19th of July, 1834, was 4'7414 when expressed in terms of one of these units the millimetre-milligramme-second unit ; was 1357 when expressed in terms of the other unit em- ployed by Gauss, and 1'357 in terms of the "unit in common use." In C.G.S. measure it would be '47414. 159. A first approximation to the distribution of mag- netic force over the earth's surface is obtained by assuming the earth to be uniformly magnetized, or, what is mathe- matically equivalent to this, by assuming the observed effects to be due to a small magnet at the earth's centre. The moment of the earth on the former supposition, or the moment of the small magnet on the latter, must be 33092 KS, K denoting the earth's radius in centims. The magnetic poles, on these suppositions, must be placed at 77 50' north lat., 296 29' east long., and at 77 50' south lat., 116 29' east long. The intensity of the horizontal component of terrestrial magnetism, at a place distant A from either of these poles, will be 33092 sin A ; 138 UNITS AND PHYSICAL CONSTANTS. [CHAP, the intensity of the vertical component will be 66184 cos A; and the tangent of the dip will be 2 cotan A. The magnetic potential, on the same supposition, will be 33092 cos A, r 2 r being variable. (See Maxwell, " Electricity and Mag- netism," vol. ii. p. 8.) Gauss's approximate expression for the potential and intensity at an arbitrary point on the earth's surface consists of four successive approximations, of which this is the first. 160. According to " Airy on Magnetism," the place of greatest horizontal intensity is in lat. long. 259 E., where the value is '3733 ; the place of greatest total in- tensity is in South Victoria, about 70 S., 160 E., where its value is '7898, and the place of least total intensity is near St. Helena, in lat. 16 S., long. 355 E., where its value is -2828. 161. The following mean values of the magnetic ele- ments at Greenwich have been kindly furnished by the Astronomer Royal (Dec., 1885) : West Declination, 18 15' '0 - (t - 1883) x 7' '80. Horizontal Force, '1809 + (t- 1883) x -00018. Dip, 67 31'-8-(- 1883) x T39. Vertical Force, 0'4374 - (t- 1883) x "00007. = Horizontal force x tan. dip. Each of these formulae gives the mean of the entire year t. 162. According to J. E. H. Gordon ("Phil. Trans.," 1877, with correction in "Proc. Roy. Soc.," 1883, pp. x.j MAGNETISM. 139 4, 5), the rotation of the plane of polarisation between two points, one centimetre apart, whose magnetic potentials (in C.G.S. measure) differ by unity, is (in circular mea- sure) 1-52381 x 10-* in bisulphide of carbon, for the principal green thallium ray, and is 2-248 x 10- in distilled water, for white light. Mr. Gordon infers from Becquerel's experiments ("Comp. Rend.," March 31, 1879) that it is about 3x 10~ 9 for coal gas. According to Lord Rayleigh (" Proc. Roy. Soc.," Dec. 29, 1884), the rotation for sodium light in bisulphide of carbon at 18 C. is -04202 minute. This is 1-22231 x 10- 5 in circular measure. 140 CHAPTER XI, P]LECTRICITY. Electrostatics. 163. IF q denote the numerical value of a quantity of electricity in electrostatic measure, the mutual force be- tween two equal quantities q at the mutual distance I will be $-. In the C.G.S. system the electrostatic unit of i~ electricity is accordingly that quantity which would repel an equal quantity at the distance of 1 centim. with a force of 1 dyne. Since the dimensions of force are , we have, as regards dimensions, q' 2 ml , ml 9 i,f - 1 ! = _-, whence f = ,q = mlt . 164. The work done in raising a quantity of electricity q through a difference of potential v is qv. Hence we have In the C.G.S. system the unit difference of potential is CHAP, xi.] ELECTRICITY. HI that difference through which a unit of electricity must be raised that the work done may be 1 erg. Or, we may define potential as the quotient of quantity of electricity by distance. This gives v = m^ftt' 1 . l~ l = mh*t~\ as before. 165. In the C.G.S. system the unit of potential is the potential due to unit quantity at the distance of 1 centim. The capacity of a conductor is the quotient of the quantity of electricity with which it is charged by the potential which this charge poduces in it. Hence we have capacity = ? = m^ftr 1 . m ~ fy ~^t = l. The same conclusion might have been deduced from the fact that the capacity of an isolated spherical con- ductor is equal (in numerical value) to its radius. The C.G.S. unit of capacity is the capacity of an isolated sphere of 1 centim. radius. 166. The numerical value of a current (or the strength of a current) is the quantity of electricity that passes in unit time. Hence the dimensions of current are ^; that is, m*ftt~*. The C.G.S. unit of current is that current which con- veys the above denned unit of quantity in 1 second. 167. The dimensions of resistance can be deduced from Ohm's law, which asserts that the resistance of a wire is the quotient of the difference of potential of its two ends, by the current which passes through it. Hence we have resistance = m^fir 1 . m ~fy~ V = T V 142 UNITS AND PHYSICAL CONSTANTS. [CHAP. Or, the resistance of a conductor is equal to the time required for the passage of a unit of electricity through it, when unit difference of potential is maintained between its ends. Hence time x potential 1 7 i -i T -4 7 -i resistance = - L__. _ = * . m 2 W" 1 . m 2 l ?t = I t. quantity 168. As the force upon a quantity q of electricity, in a field of electrical force of intensity i, is iq, we have The quantity here denoted by i is commonly called the <c electrical force at a point." Electromagnetics. 169. A current C (or a current of strength C) flowing along a circular arc, produces at the centre of the circle an intensity of magnetic field equal to C multiplied by length of arc divided by square of radius. Hence C divided by a length is equal to a field-intensity, or G = length x intensity = L . M^L " ^T" 1 = lAlV 1 . 170. The quantity of electricity Q conveyed by a cur- rent is the product of the current by the time that it lasts. The dimensions of Q are therefore L*M 2 . 171. The work done in urging a quantity Q through a circuit, by an electromotive force E, is EQ ; and the work done in urging a quantity Q through a conductor, by means of a difference of potential E between its ends, is EQ. Hence the dimensions of electromotive force, and also the dimensions of potential, are ML 2 T~ 2 . L " ^M ~ 2 , or ELECTRICITY. 143 172. The capacity of a conductor is the quotient of quantity of electricity by potential. Its dimensions are therefore M*L* . M ~ *L ~ T 2 ; that is, L^T 2 . -p 173. Resistance is .-; its dimensions are therefore M*L*T- a . M ~ *L ~ *T ; that is, LT' 1 . 174. The following table exhibits the dimensions of each electrical element in the two systems, together with their ratios : Dimensions in electrostatic system. Dimensions in electromagnetic system. Dimensions in E.S. Dimensions in E.M. Quantity, M*LT- M*L* LT- 1 Current, M*L*T-> M*L*T-> LT- 1 Capacity, Potential and ) electronic- > tive force, \ L L -1 T 2 L 2 T- 2 Resistance, L-T LT- 1 L- 2 T 2 175. The heat generated in time T by the passage of a current C through a wire of resistance E, (when no other p2"R r r work is done by the current in the wire) is gramme J degrees, J denoting 4 '2 x 10 7 ; and this is true whether C and R are expressed in electromagnetic or in electrostatic units. 144 UNITS AND PHYSICAL CONSTANTS. [CHAP. Ratios of the two sets of Electric Units. 176. Let us consider any general system of units based on a unit of length equal to L centims., a unit of mass equal to M grammes, a unit of time equal to T seconds. Then we shall have the electrostatic unit of quantity equal to M^L^T' 1 C.G.S. electrostatic units of quantity, and the electromagnetic unit of quantity equal to M 5 L 2 C.G.S. electromagnetic units of quantity. It is possible so to select L and T that the electrostatic unit of quantity shall be equal to the electromagnetic unit. We shall then have (dividing out by M^lJ) LT- 1 C.G.S. electrostatic units = 1 C.G.S. electromagnetic unit; or the ratio of the C.G.S. electromagnetic unit to the C.G.S. electrostatic unit is . Now is clearly the value in centims. per second of that velocity which would be denoted by unity in the new system. This is a definite concrete velocity ; and its numerical value will always be equal to the ratio of the electromagnetic to the electrostatic unit of quantity, whatever units of length, mass, and time are employed. 177. It will be observed that the ratio of the two units of quantity is the inverse ratio of their dimensions ; and XL] ELECTRICITY. 145 the same can be proved in the same way of the other four electrical elements. The last column of the above table shows that M does not enter into any of the ratios, and that L and T enter with equal and opposite indices, showing that all the ratios depend only on the velocity -. Thus, if the concrete velocity be a velocity of v centims. per second, the following relations will subsist between the C.G.S. units : 1 electromagnetic unit of quantity = v electrostatic units. 1 current =v 1 ,, capacity =v 2 v electromagnetic units of potential = 1 electrostatic unit. v 2 resistance = 1 ,, 178. Weber and Kohlrausch, by an experimental comparison of the two units of quantity, determined the value of v to be 3 '1074 x 10 10 centims. per second. Sir. W. Thomson, by an experimental comparison of the two units of potential, determined the value of v to be 2-825 x 10 10 . Professor Clerk Maxwell, by an experiment in which an electrostatic attraction was balanced by an electro- dynamic repulsion, determined the value of v to be 2-88 x 10 10 . Professors Ayrton and Perry, by measuring the capacity K 146 UNITS AND PHYSICAL CONSTANTS. [CHAP. of an air-condenser both electromagnetically and statically ("Nature," Aug. 29, 1878, p. 470), obtained the value 2-98 x 10 10 . Professor J. J. Thomson (" Phil. Trans.," 1883, June 21), by comparing the electrostatic and electromagnetic mea- sures of the capacity of a condenser, and employing Lord Rayleigh's latest value of the B.A. resistance coils, de- termined v to be 2-963 xlO 10 . All these values agree closely with the velocity of light in vacuo, of which the best determinations are, some of them a little less, and some a little greater than 3 x 10 10 . We shall adopt this round number as the value of v. 179. The dimensions of the electric units are rather simpler when expressed in terms of length, density, and time. Putting D for density, we have M = L 3 D. Making this substitution for M, in the expressions above obtained { 174), we have the following results : Electrostatic. . Electromagnetic. Quantity, D^T- 1 D*L 2 Current, D*L 3 T- 2 D^T- 1 Capacity, L L~ 1 T* Potential, D^T- 1 D*L 3 T-2 Resistance, L~ X T LT- 1 It will be noted that the exponents of L and T in these expressions are free from fractions. XT.] ELECTRICITY. 147 Specific Inductive Capacity. 180. The specific inductive capacity of an insulating substance is the ratio of the capacity of a condenser in which this substance is the dielectric to that of a conden- ser in other respects equal and similar in which air is the dielectric. It is of zero dimensions, and its value exceeds unity for all solid and liquid insulators. According to Maxwell's electro-magnetic theory of light, the square root of the specific indue tive capacity is equal to the index of refraction for the rays of longest wave- length. Messrs. Gibson and Barclay, by experiments performed in Sir W. Thomson's laboratory ("Phil. Trans.," 1871, p. 573), determined the specific inductive capacity of solid paraffin to be 1-977. Dr. J. Hopkinson (" Phil. Trans.," 1877, p. 23) gives the following results of his experiments on different kinds of flint glass : Kind of Flint Glass. Density. Specific Inductive Capacity. Quotient by Density. Index of Refraction for D line. Very light, ... Light, 2-87 3-2 6-57 6-85 2'29 2-14 1-541 1-574 Dense, 3 '66 7 '4 2'02 1-622 Double extra ) dense, \ 4-5 10-1 2-25 1-710 In a later series of experiments ("Phil. Trans.," 1881, Dec. 16), Dr. Hopkinson obtains the following mean determinations : 148 UNITS AND PHYSICAL CONSTANTS. [CHAP. Specific Specific Glass. Inductive Density. . Inductive Capacity. Capacity. Hardcrown, ............ 6'96 2'485 Paraffin, 2 '29 Very light flint, ......... 6'61 2'87 Light flint, .............. 6'72 3'2 Dense flint, ............... 7'38 3'66 Double extra-dense flint, 9 "90 4 '5 Plate, ..................... 8-45 181. For liquids Dr. Hopkinson ("Proc. Roy. Soc.," Jan. 27, 1881) gives the following values of /r^ (computed) and K (observed), K denoting the specific inductive capacity and /A W the index of refraction for very long waves deduced by the formula where 6 is a constant. & K Petroleum spirit (Field's), ........ ....... 1 '922 1 '92 Petroleum oil (Field's), .................. 2'075 2'07 ,, (common), .................. 2'078 2'10 Ozokerit lubricating oil (Field's), ...... 2'086 2'13 Turpentine (commercial), .............. 2'128 2'23 Castor oil, .................................... 2-153 4'78 Sperm oil, ................................... 2'135 3 '02 Olive oil, .................................... 2-131 3'16 Neatsfoot oil, .............. '. .................. 2-125 3'07 This list shows that the equality of // to K (which Maxwell's theory requires) holds nearly true for hydro- carbons, but not for animal and vegetable oils. 182. Wiillner (" Sitzungsber. konigl. bayer. Akad.," March, 1877) finds the following values of specific inductive capacity : Paraffin, ...... 1'96 Shellac, ... 2-95 to 373 Ebonite, ...... 2-56 Sulphur, ... 2'88 to 3'21 Plate glass, ... 6 -10 XL] ELECTRICITY. 149 Boltzmann ("Carl's Repertorium," x. 92165) finds the following values : Paraffin, 2'32 Colophonium, ... 2'55 Ebonite, 3'15 Sulphur, 3'84 Schiller ("Pogg. Ann.," clii. 535, 1874) finds: Paraffin. ... 5 "83 to 2 '47 Caoutchouc, ... 2'12 to 2'34 Ebonite^ ... 2'21 to 2'76 Do., vulcanized, 2 '69 to 2 '94 Plate glass, 5 '83 to 6 '34 Silow (" Pogg. Ann.," clvi. and clviii.) finds the following values for liquids : Oil of turpentine, 2-155 to 2'221 Benzene, 2'199 Petroleum, 2*039 to 2'07l Boltzmann ("Wien. Akad. Ber." (2), Ixx. 342, 1874) finds for sulphur in directions parallel to the three princi- pal axes, the values 4-773, 3-970, 3-811. 183. Quincke (" Sitz. Pretiss. Akad.," Berlin, 1883) has made the following determinations. To explain the last two columns it is to be observed that, according to Max- well's theory, the charging of a condenser produces tension (or diminution of pressure) in the dielectric along the lines of force, and repulsion (or increase of pressure) perpen- dicular to the lines of force, the tension and the repulsion being each equal to K(A-B) 2 " 87TC 2 where K denotes the specific inductive capacity of the dielectric, c the distance between the two parallel plates of the condenser, and A - B their difference of potentials. Quincke observed the tension and repulsion, and computed 150 UNITS AND PHYSICAL CONSTANTS. [CHAP. K from each of them separately. The results are given in the last two columns, and are in every case greater than the " observed " value of K obtained in the usual way by comparison of capacities. The temperature printed below the index of refraction is the temperature at which the electrical experiments were performed. Density Index of Specific Computed aiid refraction inductive / \ tempera- and tern- capacity from from ture. perature. observed, tension, repulsion. I 1-7205 1-3605 3-364 4-851 4-672 Ether, j at 14 -9 6 -60 J 1-3594 3-322 4-623 4-660 " I 8 -37 5 vols. ether to 1 bisul- ( 8134 1 -4044 2-871 4-136 4-392 phide of carbon, / 16 -4 8 -50 1 ether to 1 bisulphide, ... j 9966 16 -6 1-4955 10 -50 2-458 3-539 3-392 1 ether to 3 bisulphide, ... j 1-1360 17'4 1-5677 5 -30 2-396 3-132 3-061 Sulphur in bisulphide of ( 1 -3623 1-6797 2-113 2-870 2-895 carbon (19 '5 per cent.) j 12-6 8 -68 Bisulphide of carbon from $ 1-2760 1-6386 2-217 2-669 2-743 Kahlbaum, j 12 -2 7 -50 Bisulphide of carbon from ( 1-2796 1 -6342 1-970 2-692 2-752 Heidelberg, j 10 -2 12-98 1 vol. bisulphide to 1 tur- ( 1-0620 1-5442 1-962 2-453 2-540 pentine, \ 17'8 10'92 Heavy benzol from ben- ( 8825 1 -5035 1-928 2-389 2-370 zoic acid, \ 15 -91 13 -20 Pure benzol from benzoic ( 8822 1-5050 2-050 2-325 2-375 acid, I 17'64 14 -40 Light benzol, j 7994 17'20 1-4535 ir-eo 1-775 2-155 2-172 Rape oil, j 9159 16 -4 1-4743 16 -41 2-443 2-385 3-296 Oil of turpentine, J 8645 1-4695 16-71 1-940 2-259 2-356 Rock oil 8028 1-4483 1-705 2-138 2-149 17'0 16 -62 XL] ELECTRICITY. 151 184. Professors Ayrton and Perry have found the following values of the specific inductive capacities of gases, air being taken as the standard : Air, 1-0000 Vacuum, 0'99S5 Carbonic acid, . . 1 "0008 Hydrogen, 0'999S Coal gas, T0004 Sulphurous acid, 1 '0037 Practical Units. 185. The unit of resistance chiefly employed by practical electricians is the Ohm, which is theoretically denned as 10 9 C.G.S. electro-magnetic units of resistance. The practical unit of electro-motive force is the Volt, which is defined as . 10 8 C.G.S. electro-magnetic units of potential. The practical unit of current is the Ampere, It is de- fined as T T o of the C.G.S. electro : magnetic unit current, or as the current produced by 1 volt through 1 ohm. The practical unit of quantity of electricity is the Coulomb. It is defined as TO of the C.G.S. electro-magnetic unit of quantity, or as the quantity conveyed by 1 ampere in 1 second. The practical unit of capacity is the Farad.* It is defined as 10~ 9 of the C.G.S. electro-magnetic unit of capacity, or as the capacity of a condenser which holds 1 coulomb when charged to 1 volt. * As the farad is much too large for practical convenience, its millionth part, called the microfarad, is practically employed, and condensers are in use having capacities of a microfarad and its decimal subdivisions. The microfarad is 10~ 15 of the C.G.S. electromagnetic unit of capacity. 152 UNITS AND PHYSICAL CONSTANTS. [CHAP. The practical unit of work employed in connection with these is the Joule. It is denned as 10 7 ergs, or as the work done in 1 second by a current of 1 ampere in flowing through a resistance of 1 ohm. The corresponding practical unit of rate of working is the Watt. It is defined as 10 7 ergs per second, or as the rate at which work is done by 1 ampere flowing through 1 ohm. 186. The standard resistance-coils originally issued in 1865 as representing what is now called the ohm, were constructed under the direction of a Committee of the British Association, and their resistance was generally called the B. A. unit. The latest and best determinations by Lord Rayleigh and others have shown that it was about 1 or, more exactly, 1 '3 per cent. too small, the actual resistance of the original B. A. coils being 987 x 10 9 C.G.S. 187. An earlier unit in use among electricians was Siemens' unit, defined as the resistance at C. of a column of pure mercury 1 metre long and 1 sq. millimetre in section. The resistance of such a column is about 943 x 10 9 C.G.S. The reciprocal of -943 is 1 -06. 188. The question of what electrical units should be adopted received great attention at the International Congress of Electricians at Paris in 1881 ; and the follow- ing resolutions were adopted : XL] ELECTRICITY. 153 Resolutions adopted by the International Congress of Electricians at the sitting of September 22nd, 1881. 1. For electrical measurements, the fundamental units, the centimetre (for length), the gramme (for mass), and the second (for time), are adopted. 2. The ohm and the volt (for practical measures of resistance and electromotive force or potential) are to keep their existing definitions, 10 9 for the ohm, and 10 s for the volt. 3. The ohm is to be represented by a column of mercury of a square millimetre section at the temperature of zero centigrade. 4. An International Commission is to be appointed to determine, for practical purposes, by fresh experiments, the length of a column of mercury of a square millimetre section which is to represent the ohm. 5. The current produced by a volt through an ohm is to be called an ampere. 6. The quantity of electricity given by an ampere in a second is to be called a coulomb. 7. The capacity defined by the condition that a coulomb charges it to the potential of a volt is to be called a farad. 189. At a subsequent International Conference at Paris in 1884, it was agreed to define the "legal ohm" as " the resistance of a column of mercury 106 centimetres long and 1 sq. millimetre in section, at the temperature of melting ice." The following summary of experimental results was laid before this Conference. The two columns of numerical values are inversely proportional, their common product being 100. One of them gives the value of Siemens' 154 UNITS AND PHYSICAL CONSTANTS. [CHAP. unit in terms of the theoretical ohm (10 9 C.G.S.), and the other gives the length of a column of pure mercury at C., 1 square millimetre in section, which has a resist- ance of 1 theoretical ohm. Siemens' Column of Year. Observer. Unit in Mercury Method. Ohms. cm. 1864. British Assoc. Com., 9539 104-83 Brit. Association. 1881. Rayleigh & Shuster, 9436 105-98 Do. 1882. Rayleigh, 9410 106-28 Do. 1882. H.Weber, 9421 106-14 Do. 1874. Kohlrausch, 9442 105-91 Weber( 1st method). 1884. Mascart, 9406 106-32 Do. 1884. Wiedemann, 9417 106-19 Do. 1878. Rowland, 9453 105-79 Kirchhoff. 1882. GlazeLrook, 9408 106-30 Do. 1884. Mascart, 9406 106-32 Do. 1884. F.Weber, 9400 105-37 Do. 1884. Roiti, 9443 105-90 Roiti. 1873. Lorenz, 9337 107-10 Lorenz. 1884. Lorenz, 9417 106-19 Do. 1883. Rayleigh, 9412 106-24 Do. 1884. Zenz, 9422 106-13 Do. 1882. Dora, 9482 105-46 Weber (damping). 1883. Wild, 9462 105-68 Do. 1884. H. F. Weber, 9500 105-26 Do. 1866. Joule, 9413 106-23 Joule. Mean, 9430 106-04 Several of the most distinguished physicists present expressed their opinion that 106"2 or 106'25 centimetres was the most probable value of the required length ; but in order to obtain unanimity it was agreed to adopt the length 106 centimetres, as above stated. 1DO. By way of assisting the memory, it is useful to remark that the numerical value of the ohm is the same XI.] ELECTRICITY. 155 as the numerical value of a velocity of one earth-quadrant per second, since the length of a quadrant of the meridian is 10 9 centims. This equality will subsist whatever funda- mental units are employed, since the dimensions of resist- ance are the same as the dimensions of velocity. No special names have as yet been assigned to any electrostatic units. Electric Spark. 191. Sir W. Thomson has observed the length of spark between two parallel conducting surfaces maintained at known differences of potential, and has computed the corresponding intensities of electric force by dividing (in each case) the difference of potential by the distance, since the variation of potential per unit distance measured in any direction is always equal to the intensity of the force in that direction. His results, as given on page 258 of " Papers on Electrostatics and Magnetism," form the first two columns of the following table : Distance between Surfaces. Intensity of force in Electi-ostatic Units. Difference of Potential between Surfaces. In Electrostatic Units. In Electromagnetic Units. 0086 267-1 2-30 6-90xl0 10 0127 257-0 3-26 9-78 0127 262-2 3-33 9-99 0190 224-2 4-26 1278 0281 200-6 5'64 16-92 040S 151-5 6-18 18-54 0563 144-1 8-11 24-33 0584 139-6 8-15 2445 0688 140-8 9-69 29-07 0904 134-9 12-20 3660 1056 132-1 13-95 41-85 1325 131-0 17-36 2-08 156 UNITS AND PHYSICAL CONSTANTS. [CHAP. The numbers in the third column are the products of those in the first and second. The numbers in the fourth column are the products of those in the third by 3 x 10 10 . 192. Dr. Warren De La Rue, and Dr. Hugo W. Miiller ("Phil. Trans.," 1877) have measured the striking dis- tance between the terminals of a battery of choride of silver cells, the number of cells being sometimes as great as 11000, and the electromotive force of each being 1*03 volt. Terminals of various forms were emplo} T ed ; and the results obtained with parallel planes as terminals have been specially revised by Dr. De La Rue for the present work. These revised results (which were obtained by graphical projection of the actual observations on a larger scale than that employed for the Paper in the Philosophi- cal Transactions) are given below, together with the data from which they were deduced : DATA. Striking Distance. In Inches. In Centims. 1200 0-012 0-0305 2400 021 0533 3600 033 0838 4800 049 1245 5880 058 1473 6960 073 1854 8040 088 2236 9540 110 2794 11000 133 3378 XL] ELECTRICITY. 157 DEDUCTIONS. Intensity of Force Electromotive Force in Volts. Striking Distance in Centirns. Volts per Centim. In C.G.S. units. Electromagnetic. Electro- static. 1000 0205 48770 4-88 x 10 12 163 2000 0430 46500 4-65 155 3000 0660 45450 4-55 152 4000 0914 43770 4-38 146 5000 1176 42510 4-25 142 6000 1473 40740 4-07 136 7000 1800 38890 3-89 130 8000 2146 37280 3-73 124 9000 2495 36070 3-6! 120 10000 2863 34920 3-49 116 11000 3245 33900 3-39 113 11330 3378 33460 3-35 112 193. The resistance of a wire (or more generally of a prism or cylinder) of given material varies directly as its length, and inversely as its cross section. It is there- fore equal to -p length section where R is a coefficient depending only on the material. R is called the specific resistance of the material. Its reciprocal is called the specific conductivity of the Jti material. R is obviously the resistance between two opposite faces of a unit cube of the substance. Hence in the C.G.S. system it is the resistance between two opposite faces of a cubic centim. (supposed to have the form of a cube). The dimensions of specific resistance are resistance x length ; that is, in electromagnetic measure, velocity x length ; that is, 158 UNITS AND PHYSICAL CONSTANTS. [CHAF. RESISTANCE. 194. The following table of specific resistances is altered from that given in former editions of this work by subtracting 1'SS per cent, from all the numbers in the column headed " Specific Resistance," this being the correc- tionrequired to reduce the resistanceof mercury from 96146, the value previously given, to 94340, which is the value resulting from the new definition of the " legalohm " : Specific Resistances in Electromagnetic Measure (at C. unless otherwise stated). I Specific Resistance. Percentage variation per degree at '20 C. Specific Gravity. Silver hard-drawn, 1579 377 10*50 CoDper, 1611 388 8-95 Gold, ,, 2114 365 19-27 Lead, pressed, 19474 387 11-391 Mercury liquid, 94340 072 13-595 Gold 2, Silver 1, hard or\ annealed, / 10781 065 15-218 Selenium at 100 C., crys-\ talline / 5-9xl0 13 1-00 Water at 22 C., 7'05xl0 10 47 ,, with -2 percent.il S0 4 4-39 47 8'3 3-26 653 20 1-41 799 n >? 35 ,, , 1-24 1-259 41 1-34 1-410 Sulphate of Zinc and Water \ ZnS0 4 + 23H 2 Oat23C.,/ 1-83 Sulph. of Copper and Water) CuS0 4 + 45 H 2 at 22 C., / 1-91 Glass at 200 C. , 2-23xl0 16 ,, 250, l-36x!0 15 300, l-45x!0 14 , 400 7-21xl0 13 Gutta Percha at 24 C. , 3'46xl0 23 oc., 6-87 x 10 24 XI.] ELECTRICITY. 159 For the authorities on which this table is based see Maxwell, "Electricity and Magnetism," vol. i., last chapter. 195. The following table of specific resistances of metals at C. is reduced from Table IX. in Jen kin's Cantor Lectures. It is based on Matthiessen's experi- ments. A deduction of 1 '88 per cent, has been made, as in the preceding table : Specific Resistance. Percentage of Variation for a degree at 20 C. Resistance in i Ohms of a Wire of 1 mm. diam. 1 m. long. Silver, annealed, , , hard-drawn, Copper, annealed, ,, hard-drawn, Gold, annealed, 1492 1620 1584 1620 2041 377 388 365 0190 0206 0202 0206 0260 ,, hard-drawn, Aluminium, annealed, Zinc, pressed, 2077 2889 5581 365 0264 0368 '0749 Platinum, annealed, Iron, annealed, 8982 9638 1144 1227 Nickel, annealed, Tin, pressed, Lead, pressed, 12358 13103 19468 365 387 1573 1668 2479 Antimony, pressed, Bismuth, pressed,. 35209 130098 389 354 4483 1 '6565 Mercury, liquid, 94340 072 1-2012 Alloy, 2 parts Platinum, 1 | part Silver, by weight, j- hard or annealed, J 2419 031 0308 German Silver, hard or an- ^ nealed, j 20763 044 2644 Alloy, 2 parts Gold, 1 Sil- ^ ver, by weight, hard or |- annealed, ... J 10779 065 1372 160 UNITS AND PHYSICAL CONSTANTS. [CHAI-. Resistances of Conductors of Telegraphic Cables per nautical mile, at 24 C., in C.G.S. units. Red Sea, 7'79x 10 9 Malta- Alexandria, mean, 3 '42 ,, Persian Gulf, mean, 6'17 ,, Second Atlantic, mean, 4'19 ,, 190. The following formulae are given by Benoit* for the ratio of the specific resistance at t C. to that at C. : Aluminium, 1 + '003S76* + '000001320*- Copper, 1 + -00367* + '000 000 5S7* 2 Iron, 1 + "004516* + '000 005 828* 2 Magnesium, 1 + -003870* + '000 000 S63* 2 Silver, 1 + '003972* + '000 000 687 * 2 Tin, 1 + -004028* + "000 005 826* 2 Mercury in glass tube, ^ apparent resistance, not M + '0008649* + '000 001 12* 2 corrected for expansion, J Adopting the formula 1 + at for the ratio of the specific resistance at t to that at 0, MM. Cailletet and Bouty ("Jour, de Phys.," July, 1885) have made the following determinations of the coefficient of variation a at very low temperatures : Range of Coefficient of Temperature. Variation. Aluminium, +28 to- 91 '00385 Copper, -23 to-123 '00423 Iron, to- 92 '0049 Magnesium, to- 88 '00390 Mercury, - 40 to - 92 '00407 Silver, +30 to -102 '00385 Tin, to- 85 '00424 The new alloy called platinoid (consisting of German silver with a little tungsten) has been found by Mr. J. T. *Benoit, "Etudes expe'rimentales sur la Resistance electrique sous I'lnfluence de la Temperature." Paris, 1873. XI.] ELECTRICITY. 161 Bottoraley ("Proc. Roy. Soc.," May 7, 1885) to have an average variation of resistance with temperature of only '022 per cent, per degree centigrade, between C. and 100 C., being about half the variation of German silver. Its specific resistance ranges in different specimens from 2-9xlO- 5 to3-7xlO- 5 C.G.S. Resistances of Liquids. 197. The following tables of specific resistances of solutions are from the experiments of Ewing and Macgregor ("Trans. Roy. Soc., Ed in.," xxvii. 1873) : Solutions at 10 C. Specific Resistance. Sulphate of Zinc, saturated, 3 "37 x 10 10 ,, ,, minimum, 2*83 ,, Sulphate of Copper, saturated, 2 '93 ,, Sulphate of Potash, ,, T66 ,, Bichromate of Potash, 2 '96 ,, The following table is for solutions of sulphate of cop- per of various strengths. The first column gives the ratio by weight of the crystals to the water in which they are dissolved : Strength. Density at 10 C. Specific Resistance. Strength. Density. Specific Resistance. 1 to 40 1-0167 16-44xl0 10 1 to 4-146 1-1386 3-5 xlO 10 30 1-0216 1348 , 4 1-1432 3-41 20 1 -0318 9-87 , 3-297 1-1679 3-17 ,, 10 1-0622 590 , 3 1-1823 3-06 ,, 7 5 1 -0858 1-1174 4-73 , 3-81 , 2-597\ saturated / 1-2051 2-93 The following table is for solutions of sulphate of zinc: Strength. Density. Specific Resistance. Strength. Density at 10. Specific Resistance. 1 to 40 1-0140 18-29xl0 10 1 to 3 1-1582 337x10 20 1-0-J78 11-11 ,, 2 1-2186 803 ,, 10 1-0540 6-38 1-5 1 "270 285 -7 1 -0760 5-08 1 1 3530 3 10 ,, 5 1-1019 4-21 752 } saturated/ 1 -4220 3-37 L 162 UNITS AND PHYSICAL CONSTANTS. [CHAP. The following table for dilute sulphuric acid is from Becker's experiments, as quoted by Jamin and Bouty, torn. iv. p. Ill : Specific Resistance Density. /^At 0. At 8. At 16. At 24. ^ MO 1-37 xlO 10 1-04 xlO 10 845 x 10 10 737 xlO 10 1-20 1-33 ,, 926 666 486 5) 1-25 1-31 896 624 434 55 1-30 1-36 ,, 94 662 472 ,, 1-40 1-69 ,, 1-30 1-05 896 ?) 1-50 2-74 2-13 1-72 1-52 J? 1-60 4-82 ,, 3-02 275 2-21 ,, 1-70 9-41 6-25 4-23 ,, 3-07 ,, Resistance of Carbons. 198. The specific resistance of Carre's electric-light carbons at 20 C. is stated to be 3-927 x 10 6 C.G.S., whence it follows that the resistance of a cylinder 1 metre long and 1 centimetre in diameter is just half an ohm. The specific resistance of Gaudin's carbons is about 8 '5 x 10 6 ,, ,, retort carbon 6*7 x 10 7 graphite from 2'4 x 10 6 to4'2 x 10 r The resistance of carbon diminishes as the temperature increases, the diminution from to 100 C. being for JL \) Carre's and for Gaudin's. The resistance of an incan- descent lamp when heated as in actual use is about half its resistance cold. XL] ELECTRICITY. 163 Resistance of the Electric Arc. 199. The difference of potentials between the two carbons of an arc lamp has been found by Ayrton and Perry ("Phil. Mag.," May, 1883) to be practically in- dependent of the strength of the current, when the dis- tance between them is kept constant. It was scarcely altered by tripling the strength of the current. The apparent resistance of the arc (including the effect of reverse electromotive force) is therefore inversely as the current. The difference of potentials was about 30 volts when the current was from 6 to 12 amperes. 200. The following approximate determinations of the resistance of water and ice at different temperatures are contained in a paper by Professors Ayrton and Perry, dated March, 1877 (" Proc. Phys. Soc., London," vol. ii. p. 178):- Temp. Specific Cent. Resistance. -12-4 2'240xl0 18 - 6-2 1-023 - 5-02 9-4S6xl0 17 - 3'5 6-428 - 3-0 5-693 , about 2-46 4'844 ,, - 1-5 3-876 ,, - 0-2 2-840 , + 0-75 1-188 ,, + 2-2 2-48 x 10 16 + 4-0 9-1 x 10 15 + 7-75 5-4 x 10 14 + 11-02 , 3-4 The values in the original are given in megohms, and we have assumed the megohm = 10 15 C.G-.S. units. 164 UNITS AND PHYSICAL CONSTANTS. [CHAP. According to F. Kohlrausch (" Wied. Ann.," xxiv. p. 48, 1885) the resistance at 18 C. of water purified by distillation in vacuo is 4 x 10 10 times that of mercury. This makes its specific resistance 3-76 x 10 15 . 201. The specific resistance of glass of various kinds at various temperatures has been determined by Mr. Thomas Gray (" Proc. Roy. Soc.," Jan. 12, 1882). The following are specimens of the results : Bohemian Glass Tubing, density 2*43. At 60 6-05 xlO- 2 At 160 2'4 x 10 19 100 2 x 10- 1 174 8-7 x 10 18 130 2 x 10 20 Thomson's Electrometer Jar (flint glass), density 3 '172. At 100 2-06 x 10 23 At 160 2'45 x 10- 1 120 4-6Sxl0 2 - 180 5-6 x 10 20 140 1-06 200 1-2 The following are all at 60 C. : Bohemian Beaker, 4 '25 x 10 22 density 2 '427 7-15 ,, 2-587 Florence Flask, 4 '69 x 10 20 2'523 Test Tube, 1'44 ,, 2435 3-50 2-44 Flint Glass Tube, 3*89 x 10 22 2753 Thomson's Electro- ^ meter Jar (flint [ 1 '02 x 10 24 3'172 glass), J 202. The following appoximate values of the specific resistance of insulators after several minutes' electrifi- cation are given in a paper by Professors Ayr ton and Perry (-'Proc. Royal Society," March 21, 1878), "On the Viscosity of Dielectrics " : XI.J ELECTRICITY. 165 Specific Temperature. Resistance. Centigrade. Authority. Ayrton and Perry. (Standard adopted by ^ Latimer Clark. Ayrton and Perry. Recent cable tests. Ayrton and Perry. Mica, 8'4xl0 22 20 Gutta-Percha, 4 '5 x 10 23 24 Shellac, 9'OxlO 24 28 Hooper's Material, 1 '5 x 10 25 24 Ebonite, 2'SxlO 25 46 Paraffin, 3'4xl0 25 46 pi f Not yet measured with accuracy, but greater ' \ than any of the above. Air, Practically infinite. 203. Particulars of Board of Trade Standard Gauge of Wires (Imperial Gauge) Nos. 4 to 20. No. Diameter. Sectional area. Sq. inches. Resistance in ohms of 1 metre length pure copper at C. Milli- metres. Thou- sandths of inch. Annealed. Hard-drawn. 4 5-89 232 04227 0005929 0006065 5 5-38 212 03530 7107 7269 6 4-88 192 02895 8638 8835 7 4-47 176 02433 001029 001053 8 4-06 160 0201 1 1248 1276 9 3-66 144 01629 1536 1571 10 3-25 128 012^7 1948 1992 11 2-95 116 01057 2364 2418 12 2-64 104 008494 2951 3019 13 234 92 006647 3757 3842 14 203 80 005026 4992 5106 15 1-83 72 004070 6142 6283 16 163 64 003216 7742 7919 17 1-42 56 002463 01020 '01043 18 1-22 48 001809 01382 01414 19 1-016 40 001256 01993 02038 20 0914 36 000917 024152 02518 ! 166 UNITS AND PHYSICAL CONSTANTS. [CHAP. The heat generated per second in 1 metre length of pure copper wire at C. is -0048(0) =-) gin. deg., and at 40 C. is -0055J J gm. deg., 0. denoting the current in amperes, and D the diameter in millimetres. 204. Resistance of 1 metre length of Wires of Imperial Gauge at C. (For copper see preceding table.) No. Iron, annealed. German Silver, either annealed or hard-drawn. Platinum, annealed. Silver, annealed. 4 003606 007768 -003361 0005583 5 4322 9311 4028 6692 6 5253 01132 4896 8184 7 6261 01349 5836 9694 8 7590 01635 7074 001175 9 9339 02012 8705 1446 10 01184 02552 -01104 1834 11 01438 03096 -01340 2226 12 01795 03867 '01673 2779 13 02285 04922 -02129 3538 14 03036 06540 -02829 4700 15 03736 08047 -03482 5784 16 04708 1014 04388 7290 17 '06204 1336 -05782 9606 18 -08405 1811 07834 01301 19 1212 2611 -1130 01876 20 -1498 3226 1396 02319 Electromotive Force. 205. The electromotive force of a Daniell's cell was found by Sir W. Thomson (p. 245 of "Papers on Electricity and Magnetism ") to be 00374 electrostatic unit, XT.] ELECTRICITY. 167 from observation of the attraction between two parallel discs connected with the opposite poles of a Daniell's battery. As 1 electrostatic unit is 3 x 10 10 electromag- netic units, this is -00374 x 3 x 10 10 = M22 x 10 s electro- magnetic units, or 1-122 volt. According to Latimer Clark's experimental determina- tions communicated to the Society of Telegraph Engineers in January, 1873, the electromotive force of a Daniell's cell with pure metals and saturated solutions, at 64 F., is 1*105 volt, and the electromotive force of a Grove's cell 1-97 volt. These must be diminished by 1 per cent, because they were deduced from the assumption that the B. A. unit of resistance was correct. They will thus be reduced to 1*094 and 1*95 volts. According to the determination of F. Kohlrausch ("Pogg. Ann.," vol. cxli. [1870], and Erganz., vol. vi. [1874], p. 35) the electromotive force of a Daniell's cell is 1-138 x 10 8 , and that of a Grove's cell 1-942 x 10 s . These must be diminished by 3 per cent., because they were deduced from the value '9717 x 10 9 for Siemens' unit which is 3 per cent, too great. They will thus be reduced to 1-104 and 1*884 volts. H. S. Carhart ("Amer. Jour. Sci. Art.," Nov. 1884) has found the following different values for the electro- motive force of a Daniell's cell according to the strength of the zinc sulphate solution : Percent. Electromotive Electromotive ^^an. force in volts. Percent. force 1 1-125 10 1-118 3 1-133 15 1-115 5 1-142 20 1-111 74 1-120 25 1-11 1 168 UNITS AND PHYSICAL CONSTANTS. [CHAP. He finds by the same method the electromotive force of Latimer Clark's standard cell to be 1*434 volt. LordRayleigh ("Phil. Trans.," June 1884, p. 452) has determined the electromotive force of a Clark cell at 15 C. to be 1-435 volt. The value formerly assigned to it was 1'457 volt, and was based on the assumption that the B. A. unit of resist- ance was correct. In a supplementary paper (Jan. 21, 1886) he gives the general result for any temperature t, 1-435(1- 0-00077(^-15)}, together with full particulars as to the precautions neces- sary for securing constancy. 206. Professors Ayrton and Perry have made deter- minations of the electromotive forces called out by the contacts, two and two, of a great number of substances measured inductively. The method of experimenting is described in the Proceedings of the Royal Society for March 21, 1878. The following abstract of their latest results was specially prepared for this work by Professor Ayrton in January, 1879 : .] ELECTRICITY. ed directly by experiment, those with an asterisk by calculation, using the t of metals, all at the same temperature, there is no electromotive force, ame of a substance are the differences of potential, in volts, between that ^ il row as the number, the two substances being in contact. Thus ' lead is ict being 0'542 volts. 3fore only commercially pure. (jnoq^ ) j 'Sui^uauiuadxa jo ounj aq:} !ye aan^'Bjaduia l j l aSfeaaAy sstug; 1 i t- (N 05 OJ 00 t t <M (N CO CO 00 1 1 1 -ureSi'Buiv * * * * O CO M< IM 2 9 ? o S ou !Z ^ -L, o C5 IO O i 1 1 s o s i -,* CO CO 05 t*** "T^ CO ^^ 1 O i CO C<J 1 1 CO QO O5 i i "-H C^ CO !> 1 i 'i i 7 "i 1 1 1 o 1 I , OJI 1C ^O ^H GO ^ ^^ Tt< TH O T* O5 CO O T* Tf< ^H O "* 1 1 1 The numbers without an asterisk were obtain well-known assumption that in a compound circu The numbers in a vertical column below the n substance and the substance in the same hoiizont positive to copper, the electromotive force of conts The metals were those of commerce, and ther O CO C<l CO O ^ ip 1 1 QO CO CO ^ l^* C< ^ !> CO O I 1 1 1 O IO OO O CO ^ OO 1 1 1 * * * * 1 ' 7 7 ' f i a- i s | | j 1 !l| i ^ C c gN g PH H N ^ PQ CONTACT DIFFERENCES OF POTENTIAL IN VOLTS. j 1 a g 6 1 H "i 3 1 1 Mercury "092 '308 "502 "156 f 01 to -17 269 285 DistiUed water, ...J depend- to 148 171 to 177 I arbon. 100 345 Alum, saturated at ) 16'5 C., I .. -127 -653 -139 246 -225 Copper sulphate solu- \ tion, specific gravity, > 1-087 at 16-6C., .... j 103 Copper sulphate, satu- ) rated at 15 C., .... f 070 .. OQ d Sea salt, specific ) gravity, 1'18 at V -475 -605 -267 -856 -334 20'5C., ) O Sal-ammoniac, satu- ) rated at 15-5 C., .. f -396 -652 -189 057 -364 CQ Zinc sulphate solu- ") tion, specific grav- > ity, 1-125 at 16 -9 C., j Zinc sulphate, satu- ) rated at 15*3 C., . . I 1 Distilled water mixed ) with 3 zinc sulphate, > saturated solution, ) o r ~j 20 Distilled water, ) *|1 1 strong sulphuric > acid, ) Dd SQ 10 Distilled water, \ 1 strong sulphuric > acid, ) about -035 -w / 5 Distilled water, ) ^ ^""S) K H acid j P 4! 1 Distilled water, ) 5 strong sulphuric > 3 to 01 .. -120 .. -25 1 -fiOO CONCEN- f TRATED.1 Sulphuric acid, -< Nitric acid depend ing on carbon 1-113 720 to 1-252 to 1-300 672 Mercurous sulphate ) paste, I Distilled water, with ) acid, j The average temperature at the time of experimenting was about 16 C. All the liquids and salts employed were chemically pure ; the solids, how- ever, were only commercially pure. Solids with Liquids and Liquid* with Liquids in Air. 1 N Amalgamated Zinc. Mercury. Distilled Water. AlumSolution .satu- rated at 16 -5 C. ft OT G^> .2 * > Zinc Sulphate Solu- tion .Specific Grav- ity 1-125 at 16 -9 C. Zinc Sulphate Solu- tion, saturated at 15-3 C. 1 Distilled Water, 3 Zinc Sulphate. Strong Nitric Acid. -105 to 100 231 -043 164 +156 -536 -014 '090 -043 095 102 -565 -435 -637 -348 -430 -2S4 -200 -095 -'444 - '102 -344 -358 -429 016 848 1-298 1-456 1-269 1-609 475 -241 - 078 Example of the above table : Lead is positive to distilled water, and the contact difference of potentials is 0'171 volt. 172 UNITS AND PHYSICAL CONSTANTS. [CHAP. The authors point out that in all these experiments the unknown electromotive forces of certain air contacts are included. From these tables we find we can build up the electro- motive forces of some well-known cells. Eor example, in a Daniell's cell there are four contact differences of potential to consider, and in a Grove's cell five, viz. : DanieWs Cell. Volts. Copper and saturated copper sulphate, +0 '070 Saturated copper sulphate and saturated zinc sulphate. - 0"095 Saturated zinc sulphate and zinc, +0'430 Zinc and copper, +0 '750 1-155 Grove's Cell Copper and platinum, + 0'238 Platinum and strong nitric acid, + 0-672 Strong nitric acid and very weak sulphuric acid, + 0*078 Very weak sulphuric acid and zinc, +0 '241 Zinc and copper, +0 '750 1-979 Thermoelectricity. 207. The electromotive force of a thermoelectric circuit is called Thermoelectric force. It is proportional ccet. par. to the number of couples. The thermoelectric force of a single couple is in the majority of cases equal to the product of two factors, one being the difference of temperature of the two junctions, and the other the difference of the thermo- electric heights of the two metals at a temperature midway between those of the junctions. The current through the hot junction is from the lower to the higher metal when their heights are measured at the mean temperature. XI.] ELECTRICITY. 173 Our convention as to sign (that is, as to up and down in speaking of thermoelectric height) is the same as that adopted by Prof. Tait, and is opposite to that adopted in the first edition of this work. We have adopted it because it leads to the rule (for the Peltier and Thomson effects) that a current running down generates heat, and a current running up consumes heat. The following table of thermoelectric heights relative to lead can be employed when the mean temperature of the two junctions does not differ much from 19 or 20 C. It is taken from Jenkin's "Electricity and Magnetism," p. 176, where it is described as being compiled from Matthiessen's experiments. We have reversed the signs to suit the above convention, and have multiplied by 100 to reduce from microvolts to C.G.S. units. Thermoelectric Heights at about 20 C. Bismuth, pressed com-) _ cfiQQ mercial wire, / Bismuth, pure pressed ) ggrjo wire, . _/" Antimony, pressed wire + 280 Silver, pure hard, + 300 Zinc, puie pressed, + 370 Copper, ffalvano-plas- ) Bismuth, crystal, axial, - 6500 , , equatorial - 4500 Cobalt, -2200 tically precipitated, / + Antimony, pressed) fift , commercial wire, ) German Silver - 1175 Arsenic . + 1356 Quicksilver, - 41 '8 Iron, pianoforte wire, + 1750 Lead, Antimony, axial, + 2260 Tin, + 10 ,, equatorial, + ^640 Copper of Commerce, ... + 10 Platinum + 90 Phosphorus, red, + 2970 Tellurium, . . +50200 Gold,.. ...+ 120 Selenium, . . . . . + 80700 208. The following table is based upon Professor Tait's thermoelectric diagram (" Trans. Roy. Soc., Edin.," vol. xxvii. 1873) joined with the assumption that a Grove's cell has electromotive force 1'97 x 10 s : 174 UNITS AND PHYSICAL CONSTANTS. [CHAP. Thermoelectric Heights at t" C. in C.G.S. units. Iron, +1734- 4'87 t Steel, + 1139- 3'28* Alloy, believed to be Platinum Iridium, + 839 at all temperatures. Alloy, Platinum 95 ; Iridium 5, + 622 - '55 t ,, 90; ,, 10, + 596- T34 85; ,, 15, + 709- '63 ,, ,, 85; ,, 15, + 577 at all temperatures. Soft Platinum, - 61- 1'lOt Alloy, platinum and nickel, + 544 - 1 *10 1 Hard Platinum, + 260- '75 t Magnesium, + 244 - '95 t German Silver, -1207- 5'l2t Cadmium, + 266+ 4'29 Zinc, + 234+ 2'40 Silver, + 214+ l'50t Gold, + 283+ T02 Copper, + 136+ "95t Lead, Tin, - 43+ -55 Aluminium, - 77+ "39 t Palladium, - 625- 3'59 Nickel to 175 C., -2204- 5'l2t ,, 250 to 310 C., -8449 + 24-U ,, from 340 C., - 307- 5'12 The lower limit of temperature for the table is - 18 C. for all the metals in the list. The upper limit is 416 C., with the following exceptions : Cadmium, 258 C.; Zinc, 373 C. ; German Silver, 175 C. Ex. 1. Required the electromotive force of a copper-iron couple, the temperatures of the junctions being C. and 100 C. We have, for iron, +1734-4 -87; copper, + 136+ -95*; iron above copper, 1598 - 5 *82 XT. 1 ELECTRICITY. 175 The electromotive force per degree is 1598-5-82x50 = 1307, and the electromotive force of the couple is 1307(100-0) = 130,700, tending from copper to iron through the hot junction. By the neutral point of two metals is meant the tem- perature at which their thermoelectric heights are equal. Ex. 2. To find the neutral point of copper and iron we have 1598-5-82^ = 0, = 275; that is, the neutral point is 275 C. When the mean of the temperatures of the junctions is below this point, the current through the warmer junction is from copper to iron. The current ceases as the mean temperature attains the neutral point, and is reversed in passing it. Ex. 3. F. Kohlrausch (" Pogg. Ann. Erganz.," vol. vi. p. 35, 1874) states that, according to his determination, the electromotive force of a couple of iron and German silver is 24 x 10 5 millimetre-milligramme-second units for 1 of difference of temperatures of the junctions, at moderate temperatures. Compare this result with the above Table at mean temperature 100. The dimensions of electromotive force are M^L^T" 2 ; hence the C.G.S. value of Kohlrausch's unit islO'MO"^ = 10~ 3 , giving 2400 as the electromotive force per degree of difference. From the above table we have Iron above German silver, 2941 + -252, which, for = 100, gives 2966 as the electromotive force per degree of difference. 176 UNITS AND PHYSICAL CONSTANTS. [CHAP. Peltier and Thomson Effects. 209. When a current is sent through a circuit com- posed of different metals, it produces in general three distinct thermal effects. 1. A generation of heat to the amount per second of C 2 R ergs, C denoting the current, and R the resistance. 2. A generation of heat or cold at the junctions. This is called the Peltier effect, and its amount per second in ergs at any one junction can be computed by multiplying the difference of thermoelectric heights at this junction by + 273 and by the current, t denoting the centigrade temperature of the junction. If the current flows down (that is from greater to less thermoelectric height) the effect is a warming ; if it flows up, the effect is a cooling. Ex. 4. Let a unit current (or a current of 10 amperes) flow through a junction of copper and iron at 100 C. The thermoelectric heights at 100 C. are Iron, 1247 Copper, 231 Iron above copper, 1016 Multiplying 1016 by 373, we have about 379,000 ergs, or - of a gramme-degree, as the Peltier effect per second. Heat of this amount will be generated if the current is from iron to copper, and will be destroyed if the current is from copper to iron. 3. A generation of heat or cold in portions of the cir- cuit consisting of a single metal in which the temperature varies from point to point. This is called the Thomson effect. Its amount per second, for any such portion of xi.] ELECTRICITY. 177 the circuit, is the difference of the thermoelectric heights of the two ends of the portion, multiplied by 273 + , where t denotes the half-sum of the centigrade tempera- tures of the ends, and by the strength of the current. The Thomson effect, like the Peltier effect, is reversed by reversing the current, and follows the same rule that heat is generated when the current is from greater to less thermoelectric height. Experiment shows that the Thomson effect is insensible in the case of lead; hence the thermoelectric height of lead must be sensibly the same at all temperatures. It is for this reason that lead is adopted, by common consent, as the zero from which thermoelectric heights are to be reckoned. Ex. 5. In an iron wire with ends at C. and 100 C., the cold end is the higher (thermoelectrically) by 4-87 x 100 that is, by 487. Multiplying this differ- ence by 273 + 1(0 + 100) or 323, we have 157300 as the Thomson effect per second for unit current. This amount of heat (in ergs) is generated in the iron when the current through it is from the cold to the hot end, and is destroyed when the current is from hot to cold. Ex. 6. In a copper wire with ends at C. and 100 C., the hot end is the higher by '95 x 100 or 95. Multiply- ing this by 323, we have 30700 (ergs) as the Thomson effect per second per unit current. This amount of heat is generated in the copper when the current through it is from hot to cold, and destroyed when the current is from cold to hot. The effect of a current from hot to cold is opposite in these two metals, because the coefficients of t in the M 178 UNITS AND PHYSICAL CONSTANTS. [CHAP. expressions for their thermoelectric heights (p. 174) have opposite signs. Relation between Thermoelectric Force and the Peltier and Thomson effects. 210. The algebraic sum of the Peltier and Thomson effects (expressed in ergs) due to unit current for one second in a closed metallic circuit, is equal to the thermoelectric force of the circuit; and the direction of this thermoelectric force is the direction of a current round the circuit which would give an excess of destruction over generation of heat (so far as these two effects are concerned). Ex. 7. In a copper-iron couple with junctions at 0. and 100 C., suppose a unit current to circulate in such a direction as to pass from copper to iron through the hot junction, and from iron to copper through the cold junction. The Peltier effect at the hot junction is a destruction of heat to the amount 1016 x 373 = 379,000 ergs. The Peltier effect at the cold junction is a generation of heat to the amount 1598 x 273 = 436,300 ergs. The Thomson effect in the iron is a destruction of heat to the amount 487 x 323 = 157,300 ergs. The Thomson effect in the copper is a destruction of heat to the amount 95 x 323 = 30,700 ergs. The total amount of destruction is 567,000, and of generation 436,300, giving upon the whole a destruction of 130,700 ergs. The electromotive force of the couple is therefore 130,700, and tends in the direction of the current here supposed. This agrees with the calculation in Example 1. XL] ELECTRICITY. 179 Electrochemical Equivalents. 211. The quantity of a given metal deposited in an electrolytic cell or dissolved in a battery cell (when there is no " local action ") depends on the quantity of electricity that passes, irrespective of the time occupied. Hence we can speak definitely of the quantity of the metal that is "equivalent to" a given quantity of electricity. By the electrochemical equivalent of a metal is meant the quantity of it that is equivalent to the unit quantity of electricity. In the C. G. S. system it is the number of grammes of the metal that are equivalent to the C.G.S. electromagnetic unit of electricity. Special attention has been paid to the electrochemical equivalent of silver, as this metal affords special facilities for accurate measurement of the deposit. The latest experiments of Lord Rayleigh and Kohlrausch agree in giving 01118 as the C.G.S. electrochemical equivalent of silver.* The number of grammes of silver deposited by 1 ampere in one hour is 01118 X T \J x 3600 = 4-025. 212. The electrochemical equivalents of the most im- portant of the elements are given in the following table. They are calculated from the chemical equivalents in the preceding column by simple proportion, taking as basis the above-named value for silver. Their reciprocals are the quantities of electricity required for depositing one * Rayleigh's determination is '0111794; Kohlrausch 's, '011183; Mascart's, '011156. See "Phil. Trans.," 1884, pp. 439, 458. 180 UNITS AND PHYSICAL CONSTANTS. [CHAP. gramme. The quantity of electricity required for deposit- ing the number of grammes stated in the column " chemical equivalents" is the same for all the elements, namely, 9634 C.G.S. units. Elements. Atomic Weight. Valency. Chemi- cal Equiva- lents. Electro- chemical equivalents orgrammes per unit of electricity. Recipro- cal or Electri- city per gramme. Electro-positive Hydrogen, Potassium, 1 39-03 23-00 196-2 1077 63-18 199-8 117-4 55-88 > 5 58-6 64-88 206-4 27-04 15-96 35-37 126-54 79-76 14-01 1 1 1 3 1 2 1 2 1 4 2 3 2 2 2 2 3 2 1 1 1 3 1 39-03 23-00 65-4 107-7 31-59 63-18 99-9 199-8 29-35 58-7 18-63 27-94 29-3 32-44 103-2 9-01 7-98 35-37 126-54 79-76 4-67 0001038 004051 002387 006789 01118 003279 006558 01037 02074 003046 006093 001934 002900 003042 003367 01071 000935 0008283 003671 013134 008279 0004847 9634 246-9 418-9 147-3 89-45 305-0 152-5 96-43 48-22 328-3 164-1 517-1 344-8 328-7 297-0 93-37 1070 1207 272-4 76-14 120'8 2063 Sodium, Gold, Silver, Copper (cupric) , ,, (cuprous), Mercury (mercuric), . . . ,, (mercurous), Tin (stannic), ,, (stannous), Iron (ferric), . ,, (ferrous), Nickel, Zinc, Lead, Aluminium, Electro-negative Oxygen, Chlorine, Iodine, Bromine, Nitrogen, To find the equivalent of 1 coulomb, divide the above electrochemical equivalents by 10. To find the number of grammes deposited per hour by 1 ampere, multiply the above electrochemical equivalents by 360. XL] ELECTRICITY. 181 213. Let the " chemical equivalents " in the above table be taken as so many grammes; then, if we denote by H the amount of heat due to the whole chemical action which takes place in a battery cell during the consumption of one equivalent of zinc, the chemical energy which runs down, namely JH ergs, must be equal (if there is no wasteful local action) to the energy of the current pro- duced. But this is the product of the quantity of electricity 9634 by the electromotive force of the cell. TTT The electromotive force is therefore equal to a( ^-- In the tables of heats of combination which are in use among chemists, the equivalent of hydrogen is taken as 2 grammes, and that of zinc as 64*88 or 65 grammes. The equivalent quantity of electricity will accordingly be 9634 x 2, and the formula to be used for calculating the TTT electromotive force of a cell will be -. 19Joo In applying this calculation to Daniell's and Grove's cells, we shall employ the following heats of combination, which are given on page 614 of Watts' "Dictionary of Chemistry," vol. vii., and are based on Julius Thomsen's observations : Zn, O, SO 3 , Aq., ............... 108,462 Cu, O, SO 3 , Aq., ............... 54,225 N 2 2 , O 3 , Aq., ............... 72,940 N 2 2 , O, Aq., .................. 36,340 In Daniell's cell, zinc is dissolved and copper is set free, we have, accordingly, H = 108,462 - 54,225 = 54,237. In Grove's cell, zinc is dissolved and nitric acid is J 82 UNITS AND PHYSICAL CONSTANTS. [CHAP. changed into nitrous acid. The thermal value of this latter change can be computed from the third and fourth data in the above list, as follows : 72,940 is the thermal value of the action in which, by the oxidation of one equivalent of N 2 2 and combination with water, two equivalents of NHO 3 (nitric acid) are produced. 36,340 is the thermal value of the action in which, by the oxidation of one equivalent of N 2 2 and combination with water, two equivalents of NHO 2 (nitrous acid) are produced. The difference 36,600 is accordingly the ther- mal value of the conversion of two equivalents of nitrous into nitric acid, and 18,300 is the value for the conversion of one equivalent. In the present case the reverse changes take place. We have, therefore, H = 108,462 - 18,300 = 90,162. Taking J as 4'2 x 10 7 , the value of ^JL wil1 be 1-182 x 10 8 for Daniell's cell. 1-985 x 10 s Grove's These are greater by from 2 to 8 per cent, than the direct determinations given in 205. 214. Examples in Electricity. 1. Two conducting spheres, each of 1 centim. radius, are placed at a distance of r centims. from centre to centre, r being a large number; and each of them is charged with an electrostatic unit of positive electricity. "With what force will they repel each other 1 Since r is large, the charge may be assumed to be uni- formly distributed over their surfaces, and the force will be the same as if the charge of each were collected at its centre. The force will therefore be - of a dyne. r 2 XL] ELECTRICITY. 183 2. Two conducting spheres, each of 1 centim. radius, placed as in the preceding question, are connected one with each pole of a Daniell's battery (the middle of the battery being to earth) by means of two very fine wires whose capacity may be neglected, so that the capacity of each sphere when thus connected is sensibly equal to unity. Of how many cells must' the battery consist that the spheres may attract each other with a force of 2 of a dyne, r being the distance between their centres in cen- tims. 1 One sphere must be charged to potential 1 and the other to potential - 1. The number of cells required is 3. How many Daniell's cells would be required to pro- duce a spark between two parallel conducting surfaces at a distance of '019 of a centim., and how many at a distance of -0086 of a centim. 1 (See 178, 184.) 4-26 11QO 2-30 AnS ' <X)S7-i- 1139; .06374 = 4. Compare the capacity denoted by 1 farad with the capacity of the earth. The capacity of the earth in static measure is equal to its radius, namely 6 '37 x 10 s . Dividing by v' 2 to reduce to magnetic measure, we have '71 x 10~ 12 , which is 1 farad multiplied by '71 x 10~ 3 , or is '00071 of a farad. A farad is therefore 1400 times the capacity of the earth. 5. Calculate the resistance of a cell consisting of a plate of zinc, A square centims. in area, and a plate of copper of the same dimensions, separated by an acid 184 UNITS AND PHYSICAL CONSTANTS. [CHAP. solution of specific resistance 10 9 , the distance between the plates being 1 centim. Ans. - , or of an ohm. A A 6. Find the heat developed in 10 minutes by the passage of a current from 10 Daniell's cells in series through a wire of resistance 10 10 (that is, 10 ohms), assuming the electromotive force of each cell to be 1*1 x 10 8 , and the resistance of each cell to be 10 9 . Here we have -Total. electromotive force = 1*1 x I 9 . Resistance in battery = 10 10 . Resistance in wire = 1 10 . Current = ^ * : * = '55 x 10' 1 = -055. Heat developed in ) = (-Q55 2 ) x IQiQ _ 7 . 2()94 wire per second j 4'2 x 10 7 Hence the heat developed in 10 minutes is 4321 '4 gramme-degrees. 7. Find the electromotive force between the wheels on opposite sides of a railway carriage travelling at the rate of 30 miles an hour on a line of the ordinary gauge [4 feet 8J inches] due to cutting the lines of force of terrestrial magnetism, the vertical intensity being -438. The electromotive force will be the product of the velocity of travelling, the distance between the rails, and the vertical intensity, that is, (44-7 x 30) (2-54 x 56-5) (-438) = 84,300 electromagnetic units. This is about r~ of XL] ELECTRICITY. 185 8. Find the electromotive force at the instant of passing the magnetic meridian, in a circular coil consisting of 300 turns of wire, revolving at the rate of 10 revolutions per second about a vertical diameter \ the diameter of the coil being 30 centims., and the horizontal intensity of terrestrial magnetism being '1794, no other magnetic influence being supposed present. Self-induction can be left out of account, because the current is a maximum. The numerical value of the lines of force which go through the coil when inclined at an angle 6 to the meridian, is the horizontal intensity multiplied by the area of the coil and by sin 6; say nH-n-a 2 sin 0, where H = -1794, a =15, and n = 3QO. The electromotive force at any instant is the rate at which this quantity increases or diminishes ; that is, nH.7ra 2 cos 6 . w, if w denote the angular velocity. At the instant of passing the meridian cos 9 is 1, and the electromotive force is 7iH7ra 2 w. With 10 revolutions per second the value of o> is 27r x 10. Hence the electromotive force is 1794 x (3142) 2 x 225 x 20 x 300 = 2-39 x 10 6 . This is about of a volt. 42 190. To investigate the magnitudes of units of length, mass, and time which will fulfil the three following conditions : 1. The acceleration due to the attraction of unit mass at unit distance shall be unity. 2. The electrostatic units shall be equal to the electro- magnetic units. 186 UNITS AND PHYSICAL CONSTANTS. [CHAP. 3. The density of water at 4 C. shall be unity. Let the 3 units required be equal respectively to L centinis., M grammes, and T seconds. We have in C.G.S. measure, for the acceleration due to attraction ( 72), acceleration = C , . mass a where C = 6 '48 x 10~ 8 (distance) 2 and in the new system we are to have acceleration = mass (distance) 2 ' Hence, by division, acceleration in C.G.S. units acceleration in new units _ Q mass in C.G.S. units (distance in new units) 2 mass in new units " (distance in C.G.S. units) 2 ' that is, ^(f, This equation expresses the first of the three conditions. The equation = v expresses the second, v denoting 3 x 10 10 . The equation M = L 3 expresses the third. Substituting L 3 for M in the first equation, we find T = A /~. Hence, from the second equation, \ ^ and from the third, XL] ELECTRICITY. 187 Introducing the actual values of C and v y we have approximately T = 3928, L= 1-178 x 10 14 , M = 1-63x10*2; that is to say, The new unit of time will be about l h 5 J m ; The new unit of length will be about 118 thousand earth quadrants ; The new unit of mass will be about 2 '6 6 x 10 14 times the earth's mass. Electrodynamics. 191. Ampere's formula for the repulsion between two elements of currents, when expressed in electromagnetic units, is cc' ds . ds' /c . . / /) /> - - -- (2 sm a sin a cos - cos a cos a ), where c, c' denote the strengths of the two currents ; ds, ds' the lengths of the two elements ; a, a! the angles which the elements make with the line joining them ; r the length of this joining line ; the angle between the plane, of r, ds } and the plane of r, ds'. For two parallel currents, one of which is of infinite length, and the other of length Z, the formula gives by integration an attraction or repulsion, where D denotes the perpendicular distance between the currents. 188 UNITS AND PHYSICAL CONSTANTS. [CHAP. xi. Exa/mple. Find the attraction between two parallel wires a metre long and a centim. apart when a current of is passing through each. Here the attraction will be sensibly the same as if one of the wires were indefinitely increased in length, and will be 200/1 , that is, each wire will be attracted or repelled with a force of 2 dynes, according as the directions of the currents are the same or opposite. 189 OMISSION (TO BE ADDED TO 63, p. 61). ACCORDING to experiments by Quincke (Berlin Transac- tions, April 5, 1885) the following are the compressions due to the pressure of one atmosphere. They are ex- pressed in millionths of the original volume : Glycerine, 25'24 Rape oil (rttbol), 48 '02 Almond oil, 48'21 Olive oil, 48-59 Water, Bisulphide of carbon, .... Oil of turpentine, Benzol from benzoic acid Benzol, Petroleum, Alcohol, Ether, ...115'57 at C. it t C. t. 25-24 25-10 19-00 48-02 58-18 17-80 48-21 56-30 19-68 48-59 61-74 18-3 50-30 45-63 22-93 53-92 63-78 17-00 58-17 77-93 18-56 66-10 16-78 62-84 16-08 64-99 74-50 19-23 82-82 95-95 17-51 115-57 147-72 21-36 CORRECTION (p. 84). BEXOIT'S results on refraction of air will not appear in vol. v., but in a later volume. 190 SUGGESTION FOR WRITING DECIMAL MULTIPLES AND SUBMULTIPLES. PROFESSOR NEWCOMB has suggested, as a possible improve- ment in future editions of this work, the employment of powers of 1000 instead of powers of 10 as factors (a plan which corresponds with the usual division of digits into periods of 3 each), and the employment of the letter m in this connection to denote 1000. Thus, instead of 1'226 x 10 5 , we should write 122-6 m. l-006xl0 7 , 10-06 m 2 . 000 000 9, -9 m~~. The plan appears to possess some advantages ; and if the symbol m for 1000 is not sufficiently self-explanatory, we might write 122'6 x 10 3 , 10'06 x 10 6 , '9 x 10~ 6 . We place the suggestion on record here that it may not be overlooked. 191 APPENDIX. First Report of the Committee for the Selection and Nomenclature of Dynamical and Electrical Units, the Committee consisting of SIR W. THOMSON, F.R.S., PROFESSOR G. C. FOSTER, F.R.S., PROFESSOR J. C. MAXWELL, F.R.S., MR. G. J. STONEY, F. R.S.,* PROFESSOR FLEEMING JENKIN, F.R.S., DR. SIEMENS, F.R.S., MR. F. J. BRAMWELL, F.R.S., and PROFESSOR EVERETT (Reporter). WE consider that the most urgent portion of the task intrusted to us is that which concerns the selection and nomenclature of units of force and energy ; and under this head we are prepared to offer a definite recommendation. A more extensive and difficult part of our duty is the selection and nomenclature of electrical and magnetic units. Under this head we are prepared with a definite recommendation as regards selection, but with only an interim recommendation as regards nomenclature. Up to the present time it has been necessary for every person who wishes to specify a magnitude in what is called " absolute " measure, to mention the three fundamental units of mass, length, and time which he has chosen as the basis of his system. This necessity will be obviated if one definite selection of three funda- mental units be made once for all, and accepted by the general consent of scientific men. We are strongly of opinion that such a selection ought at once to be made, and to be so made that there will be no subsequent necessity for amending it. We think that, in the selection of each kind of derived unit, all arbitrary multiplications and divisions by powers of ten, or other factors, must be rigorously avoided, and the whole system of * Mr. Stoney objected to the selection of the centimetre as the unit of length. 192 APPENDIX. fundamental units of force, work, electrostatic, and electromag- netic elements must be fixed at one common level that level, namely, which is determined by direct derivation from the three fundamental units once for all selected. The carrying out of this resolution involves the adoption of some units which are excessively large or excessively small in comparison with the magnitudes which occur in practice ; but a remedy for this inconvenience is provided by a method of denoting decimal multiples and sub-multiples, which has already been extensively adopted, and which we desire to recommend for general use. On the initial question of the particular units of mass, length, and time to be recommended as the basis of the whole system, a protracted discussion has been carried on, the principal point discussed being the claims of the gramme, the metre, and the second, as against the gramme, the centimetre, and the second, the former combination having an advantage as regards the simplicity of the name metre, while the latter combination has the advantage of making the unit of mass practically identical with the mass of unit-volume of water in other words, of making the value of the density of water practically equal to unity. We are now all but unanimous in regarding this latter element of simplicity as the more important of the two ; and in support of this view we desire to quote the authority of Sir W. Thomson, who has for a long time insisted very strongly upon the necessity of employing units which conform to this condition. We accordingly recommend the general adoption of the Centi- metre, the Gramme, and the Second as the three fundamental units ; and until such time as special names shall be appropriated to the units of electrical and magnetic magnitude hence derived, we recommend that they be distinguished from "absolute" units otherwise derived, by the letters "C.G.S. " prefixed, these being the initial letters of the names of the three fundamental units. Special names, if short and suitable, would, in the opinion of a majority of us, be better than the provisional designations "C.G.S. unit of . . . ." Several lists of names have already been suggested ; and attentive consideration will be given to any further APPENDIX. 193 suggestions which we may receive from persons interested in electrical nomenclature. The "ohm," as represented by the original standard coil, is approximately 10 9 C.G.S. units of resistance; the "volt" is approximately 10 8 C.G.S. units of electromotive force ; and the "farad" is approximately - of the C.G.S. unit of capacity. For the expression of high decimal multiples and sub-multiples, we recommend the system introduced by Mr. Stoney, a system which has already been extensively employed for electrical pur- poses. It consists in denoting the exponent of the power of 10, which serves as multiplier, by an appended cardinal num- ber, if the exponent be positive, and by a prefixed ordinal number if the exponent be negative. Thus 10 9 grammes constitute a gramme-nine; - Q of a gramme constitutes a ninth-gramme; the approximate length of a quadrant of one of the earth's meridians is a metre-seven, or a centimetre- nine. For multiplication or division by a million, the prefixes mega * and micro may conveniently be employed, according to the present custom of electricians. Thus the megohm is a million ohms, and the microfarad is the millionth part of a farad. The prefix mega is equivalent to the affix six. The prefix micro is equivalent to the prefix sixth. The prefixes kilo, hecto, deca, deci, centi, milli can also be em- ployed in their usual senses before all new names of units. As regards the name to be given to the C.G.S. unit of force, we recommend that it be a derivative of the Greek 8tva.fjus. The form dynamy appears to be the most satisfactory to etymologists. Dynam is equally intelligible, but awkward in sound to English ears. The shorter form, dyne, though not fashioned according to strict rules of etymology, will probably be generally preferred in this country. Bearing in mind that it is desirable to construct a system with a view to its becoming international, we think that * Before a vowel, either meg or megal, as euphony may suggest, may be employed instead of mega. 194 APPENDIX. the termination of the word should for the present be left an open question. But we would earnestly request that, whichever form of the word be employed, its meaning be strictly limited to the unit of force of the C.G.S. system that is to say, the force which, acting upon a gramme of matter for a second, generates a velocity of a centimetre per second. The C.G.S. unit of work is the work done by this force working through a centimetre; and we propose to denote it by some deriva- tive of the Greek Zpyov. The forms ergon, ergal, and erg have been suggested; but the second of these has been used in a different sense by Clausius. In this case also we propose, for the present, to leave the termination unsettled ; and we request that the word ergon, or erg, be strictly limited to the C.G.S. unit of work, or what is, for purposes of measurement, equivalent to this, the C.G.S. unit of energy, energy being measured by the amount of work which it represents. The C.G.S. unit of power is the power of doing work at the rate of one erg per second ; and the power of an engine, under given conditions of working, can be specified in ergs per second, For rough comparison with the vulgar (and variable) units based on terrestrial gravitation, the following statement will be useful : The weight of a gramme, at any part of the earth's surface, is about 980 dynes, or rather less than a kilodyne. The weight of a kilogramme is rather less than a megadyne, being about 980,000 dynes. Conversely, the dyne is about 1 '02 times the weight of a milli- gramme at any part of the earth's surface ; and the megadyne is about 1 '02 times the weight of a kilogramme. The kilogrammetre is rather less than the ergon-eight, being about 98 million ergs. The gramme-centimetre is rather less than the kilerg, being about 980 ergs. For exact comparison, the value of g (the acceleration of a body falling in vacuo) at the station considered must of course be known. In the above comparison it is taken as 980 C.G.S. units of acceleration. APPENDIX. 195 One Iwrse-power is about three quarters of an erg-ten per second. More nearly, it is 7 '46 erg -nines per second, and one force-de-cheval is 7*36 erg-nines per second. The mechanical equivalent of one gramme-degree (Centigrade) of heat is 41'6 megalergs, or 41,600,000 ergs. Second Report of the Committee for the Selection and Nomenclature of Dynamical and Electrical Units, the Committee consisting of PROFESSOR SIR W. THOMSON, F.R.S., PROFESSOR G. C. FOSTER, F.R.S., PROFESSOR J. CLERK MAXWELL, F.R.S., G. J. STONEY, F.R.S., PROFESSOR FLEEMING JENKIN, F.R.S., DR. C. W. SIEMENS, F.R.S., F. J. BRAMWELL, F.R.S., PROFESSOR W. G. ADAMS, F.R.S., PROFESSOR BALFOUR STEWART, F.R.S., and PROFESSOR EVERETT (Secretary). THE Committee on the Nomenclature of Dynamical and Electrical Units have circulated numerous copies of their last year's Report among scientific men both at home and abroad. They believe, however, that, in order to render their recom- mendations fully available for science teaching and scientific work, a full and popular exposition of the whole subject of physical units is necessary, together with a collection of examples (tabular and otherwise) illustrating the application of systematic units to a variety of physical measurements. Students usually find peculiar difficulty in questions relating to units ; and even the experienced scientific calculator is glad to have before him con- crete examples with which to compare his own results, as a security against misapprehension or mistake. Some members of the Committee have been preparing a small volume of illustrations of the C.G.S. system [Centimetre-Gramme- Second system] intended to meet this want. [The first edition of the present work is the volume of illustra- tions here referred to.] 196 INDEX. The numbers refer to the pages. Acceleration, 25. Acoustics, 70-74. Adiabatic compression, 125. Air, collected data for, 129. , density of, 43. , expansion of, 99. , specific heat of, 94, 123. , thermal conductivity of, 108. Ampere as unit, 151-153. Ampere's formula, 187. Aqueous vapour, pressure of, 100-102. , density of, 102. Astronomy, 65-69. Atmosphere, standard, 42, 43. , its density upwards, 47. Atomic weights, 180. Attraction, constant of, 67. at a point, 17. Angle, 16. , solid, 17. Barometer, correction for capil- larity, 51. Barometric measurements of heights, 47. pressure, 42. Batteries, 166-168, 172, 181. Boiling points, 98. of water, 100-102. Boyle's law, departures from, 99. Bullet, melted by impact, 31. Candle, standard, 86. Capacity, electrical, 141-143. , specific inductive, 147-150. , thermal, 87-95. Capillarity, 49-51. Carcel, 86. Cells, 166-168, 172, 181. Centimetre, reason for selecting, 23, 192. Centre of attraction, strength of, 17. Centrifugal force, 32. at equator, 34. C.G.S. system, 23, 192. Change of volume in evapora- tion, 97. in melting, 96, 97. Change-ratio, 9. Chemical action, heat of, 122. equivalents, 180. Clark's standard cell, 168. Cobalt, magnetization of, 136. Coil, revolving, 185. Combination, heat of, 122, 181. Combustion, heat of, 122. Common scale needed, 22. omparisoiiof standards (French and English), 1, 2. INDEX. 197 Compressibility of liquids, 60, 61, 189. of solids, 61-63. Compression, adiabatic, 125. Conductivity (thermal) defined, 103. , thermometric and calori- metric, 105. of air, 108. of liquids, 116, 117. of various solids, 109-116. Congress of electricians, 153. Contact electricity, 168-172. Cooling, 117-120. Current, heat generated by, 143, 166. , unit of, 141, 142, 151, 153. Curvature, dimensions of, 17, 18. Daniell's cell, 166, 167, 173, 181. Day, sidereal, 66. Decimal multiples, 24, 190, 193. Declination, magnetic, at Green- wich, 138. Densities, table of, 40. of gases, 44. of water, 38-39. Density as a fundamental unit, 146. Derived units, 5, 6. Dew-point from wet and dry bulb, 102. Diamagnetic substances, 133. Diamond, specific heat of, 90. Diffusion, coefficient of, 105-108. Diffusivity (thermal), 105. Dimensional equations, 9, 34-37. Dimensions, 7-9, 34-37. Dip at Greenwich, 138. Dispersive powers of gases, 83- 85. of solids and liquids, 77-82. Diversity of scales, 22. "Division," extended sense of, 10. Double refraction, 81. Dynamics, 15-17. Dyne, 27, 193. Earth as a magnet, 136. , size, figure, and mass of, 65. Elasticity, 52-64. , effected by heat of com- pression, 127. Electric units, tables of their dimensions, 143, 146. Electricity, 140-148. Electrochemical equivalents, 180. Electrodynamics, 187. Electromagnetic units, 142. Electromotive force, 166-172, 180-182. Electrostatic units, 140. Emission of heat, 117-120. Energy, 29. , dimensions of, 16. Equations, dimensional, 9, 34-37. , physical, 12. Equivalent, mechanical, of heat, 120. Equivalents, electrochemical, 180. Erg, 29, 194. Evaporation, change of volume in, 97. Examples in electricity, 182-1 86. in theory of units, 12-15, 34-37. in magnetism, 134-137. Expansion of gases, 99. of mercury, 128, 129. of various substances, 128, 129. Extended sense of "multiplica- tion" and "division," 10. Farad, 151-153. compared with earth, 183. Field, intensity of, 131. Films, tension in, 49, 50. , thickness of, 50, 51. 198 INDEX. Foot-pound and foot-poundal, Force, 27. , dimensions of, 15. at a point, 17. , various units of, 4. Freezing-point, change with pressure, 124. Frequencies of luminous vibra- tions, 77. Fundamental units, 6. , choice of, 19. reduced to two, 68. Gases, densities of, 44. , expansion of, 99. , indices of refraction of, 82. , inductive capacities of, 151. , two specific heats of, 123. Gauss's expression for magnetic potential, 138. pound-magnet, 134. units of intensity, 137. Geometrical quantities, dimen- sions of, 15-18. Gottingen, total intensity at, 137. Gramme-degree (unit of heat), 88. Gravitation in astronomy, 67. Gravitation measure of force and work, 28, 30. Gravity, terrestrial, 25-27. Greenwich, magnetic elements at, 138. Grove's cell, 167, 172, 181. Heat, 87-130. generated by current, 143, 166. , mechanical equivalent of, 120. of combination, 122, 181. of compression, 125. , unit of, 87, 88. , various units of, 3. Height, measured by barometer, 47. Homogeneous atmosphere, 45-47. Horse-power, 30. Hydrostatics, 38-51. Hypsometric table of boiling points, 100. Ice, specific gravity of, 96, 125. , specific heat of, 92. , electrical resistance of, 163. Indices of refraction, 77-85. related to induc- tive capacities, 147, 148. Inductive capacity, 147-150. Induction, magnetic, coefficient of, 133. Insulators, resistance of, 164, 165. Interdiffusion, 106-108. Joule's equivalent, 120. Kilogramme and pound, 2. Kinetic energy, 29. Kupffer's determination of den- sity of water, 38. Large numbers, mode of expres- sing, 24, 190, 193. Latent heats, 95-98. Latimer Clark's cell, 168. Light, 75-86. , velocity of, 75, 76. , wave-lengths of, 76. Magnetic elements at Green- wich, 138. susceptibility, 133. - units, 131, 132. Magnetism, 131-139. , terrestrial, 136-138. Magneto-optic rotation, 139. Magnetization, intensity of, 132, 133. Mass, standards of, 20. INDEX. 199 Mechanical equivalent of heat, 120. quantities, dimensions of, 15. units, 27. Mega, as prefix, 42, 193. Melting points, 95-97. Metre and yard, 1. Micro as prefix, 193. Microfarad, 151. Moment of couple, 16. of inertia, 16. of magnet, 132. of momentum, 16. Momentum, 15. Moon, 66. ' 'Multiplication, "extended sense of, 10. Neutral point (thermoelectric), 175. Newcomb on decimal multiples, 190. Nickel, magnetization of, 136. Numerical value, 5. Ohm as unit, 151-154. earth quadrant per second, 155. , "legal," 153. Optics, 75-86. Paramagnetic substances, 133. Pendulum, seconds', 25, 26. " Per," meaning of, 10. Physical deductions from di- mensions, 34-37. Platinoid, 160. Platinum, specific heat of, 90. Poisson's ratio, 62. Potential, electric, 140. , magnetic, 131. Poundal, 28. Powers of ten as factors, 24, 190, 193. Pressure, dimensions of, 17. of liquid columns, 42. Pressure, various units of, 3, 4. Pressures of vapours, 101. Pressure- height, 46. Quantity of electricity, 140, 142. Radian, 16. Radiation, 117-120. Ratios of two sets of electric units, 143. Refraction, indices of, 77-85. Reports of Units Committee, 191-195. Resilience, 54. affected by heat of com- pression, 126. Resistance, electrical, 158-166. of a cell, 161, 183. of wires, 165, 166. Rigidity, simple, 55. Rotating coil, 185. Saturation, magnetic, 133-136. Shear, 55-58. Shearing stress, 58-60. Siemens' unit, 152-154. Soap films, 50. Sound, faintest, 74. , velocity of, 70-73. Spark, length of, 155-157. Specific gravities, 40. , heat, 88-95. , two, of gases, 123. , inductive capacity, 147- 150. Spring balance, 31. Standards, French and English, 1,2. of length, 21. of mass, 20. of time, 21. Steam, pressure and density of, 100-102. , total and latent heat of, 98. Stoney's nomenclature for multiples, 193. 200 INDEX. Strain, 52, 53, 55-58. , dimensions of, 53. Stress, 52-54, 58-60. , dimensions of, 54. Strings, musical, 73. Sun's distance and parallax, 66. Supplemental section on dimen- sions, 34-37. Surface-conduction, 117-120. Surface-tension, 49, 50. Telegraphic cables, resistance of, 160. Tenacities, table of, 64. Tensions of liquid surfaces, 49, 50. Thermodynamics, 120-128. Thermoelectricity, 172-178. Time, standard of, 21. Tortuosity, 17. Two fundamental units suffici- ent, 68. Unit, 5. Units, derived, 5, 6. , dimensions of, 7-9. , special problems on, 69, 185. Vapours, pressure of, 101. Velocity, 6, 9. of light, 75, 76. of sound, 70-73. , various 'units of, 2. Vibrations per second of light, 77. Volt, 151-153. Volume, by weighing in water, 40. of a gramme of gas, 44. , unit of, 5. Volume resilience, 55, 60-63, 189. Water, compressibility of, 60, 61, 189. , density of, 38, 39. , expansion of, 39. , specific heat of, 87, 88. , weighing in, 40. | Watt (rate of working), 4, 30. Weight, force, and mass, 27, 28. , standards of, 20-2. Wires (Imperial gauge), 165, 166. Work, 29, 30, 3, 4. , dimensions of, 16. done by current, 143. Working, rate of, 30, 3, 4. Year, sidereal and tropical, 6 Young's modulus, 55. S . 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